Progressions and logarithms

In short

A progression is a sequence whose terms follow one fixed rule — a constant difference (AP), a constant ratio (GP) or a constant difference between reciprocals (HP) — and the matching *series* is the sum of those terms. A logarithm is the inverse of exponentiation: log₍a₎N is the power to which the base a must be raised to give N, which turns multiplication into addition and powers into multiplication. Together they cover growth, decay, instalments, depreciation, counting digits and comparing very large numbers. They are standard marks in first-year engineering mathematics (AICTE/CBCS "Engineering Mathematics-I", sequences and series) and in GATE aptitude, and they appear in almost every placement aptitude paper — TCS NQT, Infosys, Wipro, Capgemini, Accenture, Cognizant — as well as CAT, bank PO, SSC and RRB exams.

Explained

Sequences, series and arithmetic progression (AP)

• Sequence — an ordered list a₁, a₂, a₃, …; series — the sum a₁ + a₂ + ⋯. A finite sequence has a last term; an infinite one does not.
• AP — each term exceeds the previous by a fixed common difference d: a, a+d, a+2d, … Here d = aₙ − aₙ₋₁ and may be negative or zero.
• nth term aₙ = a + (n−1)d. nth term from the end = l − (n−1)d, where l is the last term.
• Sum Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l) — that is, n × (average of first and last term). Also aₙ = Sₙ − Sₙ₋₁.
• Number of terms n = [(l − a)/d] + 1 — the standard way to count multiples in a range (e.g. three-digit multiples of 7).
• Useful properties: adding/subtracting a constant to every term, or multiplying every term by a constant, keeps it an AP (d changes in the second case). Terms equidistant from the two ends have a constant sum: a₁ + aₙ = a₂ + aₙ₋₁.
• Three terms in AP are best taken as a−d, a, a+d (sum = 3a); four as a−3d, a−d, a+d, a+3d (sum = 4a) — this kills one unknown immediately.
• If a, b, c are in AP then 2b = a + c; b is the arithmetic mean.
• n arithmetic means inserted between a and b form an AP of (n+2) terms with common difference d = (b − a)/(n + 1); their sum = n(a + b)/2.
• An AP's graph is a straight line in n; Sₙ is a quadratic in n of the form An² + Bn with A = d/2.

Geometric progression (GP)

• GP — each term is the previous one multiplied by a fixed common ratio r (r ≠ 0): a, ar, ar², … Here r = aₙ/aₙ₋₁.
• nth term aₙ = arⁿ⁻¹; nth term from the end = l/rⁿ⁻¹.
• Sum of n terms Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1 (use a(1 − rⁿ)/(1 − r) when r < 1 to avoid negative signs); Sₙ = na when r = 1.
• Infinite GP converges only when |r| < 1, giving S∞ = a/(1 − r). If |r| ≥ 1 the sum diverges — a favourite trap.
• If a, b, c are in GP then b² = ac; b is the geometric mean √(ac).
• n geometric means between a and b: common ratio r = (b/a)^1/(n+1); their product = (ab)^n/2.
• Properties: every term multiplied by a constant is still a GP with the same r; reciprocals of a GP form a GP with ratio 1/r; squares of a GP form a GP with ratio r².
• Recurring decimals are infinite GPs, which is why 0.4̄5̄ = 45/99 = 5/11.
• Growth and decay: r > 1 is compound growth (CI, population, revenue); 0 < r < 1 is decay (written-down-value depreciation, half-life, rebound heights).

Harmonic progression, means and the AM–GM–HM inequality

• HP — a sequence whose reciprocals are in AP. nth term of the HP = 1/[a + (n−1)d], where a and d belong to the reciprocal AP. There is no simple closed-form sum for an HP in placement syllabi; always convert to the AP of reciprocals.
• If a, b, c are in HP then 2/b = 1/a + 1/c, i.e. b = 2ac/(a + c) — the harmonic mean.
• Means of two positive numbers: A = (a+b)/2, G = √(ab), H = 2ab/(a+b), and G² = A·H (so G is the geometric mean of A and H).
• AM ≥ GM ≥ HM for positive numbers, with equality only when all the numbers are equal. This single inequality solves most "find the minimum value" aptitude questions.
• Consequences used directly: x + 1/x ≥ 2 for x > 0; for a fixed sum the product is largest when the numbers are equal; for a fixed product the sum is smallest when they are equal.
• Where each mean belongs: AM for simple averages, GM for average *rates* of growth (CAGR, index numbers), HM for average speed over equal distances and for rates like items per hour.

Special series and the arithmetico-geometric progression (AGP)

• Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]² = (Σn)² — standard results, derivable by the method of differences or induction (both examined).
• Sum of first n odd numbers = n²; sum of first n even numbers = n(n+1).
• AGP — term-by-term product of an AP and a GP: a, (a+d)r, (a+2d)r², … Its finite sum is found by multiplying by r, subtracting, and summing the resulting GP. For |r| < 1, S∞ = a/(1−r) + dr/(1−r)².
• Method of differences: if the differences of a sequence form an AP or GP, the nth term is got by summing those differences — used for series like 2, 6, 12, 20, … (nth term n(n+1)).
• Telescoping: 1/[n(n+1)] = 1/n − 1/(n+1), so Σ from 1 to n = 1 − 1/(n+1) = n/(n+1).
• Convergence (GATE/CBCS): a geometric series Σarⁿ converges iff |r| < 1; Σ1/n^p converges iff p > 1 (so the harmonic series Σ1/n diverges although its terms tend to 0). The ratio test uses lim |aₙ₊₁/aₙ|.

Logarithms: definition and laws

• Definition: if aˣ = N then x = log₍a₎N. Required conditions: N > 0, a > 0, a ≠ 1. The log of zero or of a negative number is undefined in the real system.
• Basic values: log₍a₎1 = 0, log₍a₎a = 1, log₍a₎aⁿ = n, and a^log₍a₎N = N.
• Laws (all for valid bases and positive arguments):
  • product: log(MN) = log M + log N
  • quotient: log(M/N) = log M − log N
  • power: log(Mᵖ) = p·log M
  • root: log(M^1/q) = (1/q)·log M
  • base of a power: log₍a^q₎(M^p) = (p/q)·log₍a₎M
• Change of base: log₍a₎N = log₍b₎N / log₍b₎a = ln N / ln a. Hence log₍a₎b · log₍b₎a = 1 and log₍a₎b · log₍b₎c · log₍c₎a = 1.
• Common log base 10 (written log, used with tables and for digit counting); natural log base e ≈ 2.71828 (written ln, used in calculus, decay and continuous compounding); base 2 (information, algorithm complexity).
• Monotonicity: log₍a₎x increases with x when a > 1 and decreases when 0 < a < 1 — so an inequality log₍a₎x > log₍a₎y reverses to x < y when the base is a proper fraction. This is the most common sign error.
• Standard values to memorise: log 2 = 0.3010, log 3 = 0.4771, log 5 = 0.6990, log 7 = 0.8451, ln 2 ≈ 0.6931, ln 10 ≈ 2.3026.
• Solving log equations: combine into a single log, convert to exponential form, then check the domain — roots that make any argument ≤ 0 are rejected.

Characteristic, mantissa and practical log work

• log N = characteristic + mantissa. The characteristic is the integral part (may be negative); the mantissa is the non-negative decimal part read from a log table.
• For N ≥ 1 with k digits before the decimal point, characteristic = k − 1 (log 7246 → characteristic 3).
• For N < 1 with k zeros immediately after the decimal point, characteristic = −(k+1), written in bar form (log 0.0025 = 3̄.3979, meaning −3 + 0.3979).
• The mantissa depends only on the digits, not on the decimal position: log 25, log 2.5 and log 0.025 share the mantissa 0.3979.
• Antilog reverses the process: antilog(2.8573) = 10^2.8573 ≈ 720 (mantissa gives the digits 7200 → 7.20, characteristic 2 places the point).
• Number of digits in a positive integer N = ⌊log₁₀N⌋ + 1 — the standard "how many digits in 2¹⁰⁰" question.
• Logarithms underlie log scales: sound intensity level = 10 log₁₀(I/I₀) and sound pressure level = 20 log₁₀(p/p₀) in decibels, pH = −log₁₀[H⁺], Richter magnitude, and semi-log/log-log plots that straighten exponential and power-law data.

Where progressions and logarithms meet money

• Compound interest is a GP: Aₙ = P(1 + i)ⁿ, where i is the rate per period. Simple interest balances form an AP.
• Written-down-value (WDV) depreciation under Schedule II of the Companies Act, 2013 is a GP: book value after n years = C(1 − d)ⁿ; straight-line depreciation is an AP.
• CAGR = (Final/Initial)^1/n − 1 — a geometric mean of growth factors, computed with logs.
• Annuity / SIP future value is the sum of a GP; EMI comes from the sum of a discounted GP: EMI = P·i(1+i)ⁿ / [(1+i)ⁿ − 1], the formula behind every bank and NBFC loan schedule in India.
• Doubling time: n = log 2 / log(1 + i); the "rule of 72" is the log approximation of it.

Key points

• AP — constant difference d; aₙ = a + (n−1)d; Sₙ = (n/2)(a + l).
• GP — constant ratio r; aₙ = arⁿ⁻¹; Sₙ = a(rⁿ − 1)/(r − 1).
• Infinite GP — S∞ = a/(1 − r) exists only if |r| < 1; otherwise the sum does not exist.
• HP — reciprocals in AP; always solve through the reciprocal AP, never directly.
• Means — A = (a+b)/2, G = √(ab), H = 2ab/(a+b); G² = AH and AM ≥ GM ≥ HM.
• Harmonic mean — the correct average for speeds over equal distances, not the arithmetic mean.
• Symmetric substitution — three terms in AP as a−d, a, a+d; three in GP as a/r, a, ar, so the sum or product gives one unknown at once.
• Counting formula — n = (l − a)/d + 1 counts terms, multiples or dates in a range.
• Special sums — Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = (Σn)².
• AGP — multiply by r and subtract; S∞ = a/(1−r) + dr/(1−r)² for |r| < 1.
• Logarithm — aˣ = N ⇔ x = log₍a₎N, with N > 0, a > 0, a ≠ 1.
• Laws — log(MN) = log M + log N; log(M/N) = log M − log N; log Mᵖ = p log M.
• Change of base — log₍a₎N = log N / log a; log₍a₎b · log₍b₎a = 1.
• Characteristic — k − 1 for a k-digit number; −(k+1) when k zeros follow the decimal point.
• Digit count — digits in N = ⌊log₁₀N⌋ + 1; log 2 = 0.3010 and log 3 = 0.4771 answer most such questions.
• Base < 1 — log₍a₎x is decreasing, so inequalities flip; a classic marks-losing slip.
• Domain check — every solution of a log equation must keep all arguments positive.
• Finance link — CI, WDV depreciation, SIP and EMI are all GPs; CAGR and doubling time need logs.

Formulas and examples

Example 1 — AP: number of terms, and counting multiples (a) How many terms of 3, 7, 11, … add up to 210?

• a = 3, d = 4, Sₙ = 210 → (n/2)[6 + (n−1)4] = 210 → n(4n + 2) = 420 → 2n² + n − 210 = 0.
• n = [−1 + √(1 + 1680)]/4 = (−1 + 41)/4 = 10 terms. (Check: 10th term = 3 + 9×4 = 39; S = (10/2)(3 + 39) = 210 ✓)

(b) How many three-digit numbers are divisible by 7?

• First = 105, last = 994, d = 7 → n = (994 − 105)/7 + 1 = 127 + 1 = 128 numbers.

Example 2 — GP: depreciation and a finite sum A CNC machine bought for ₹8,00,000 is depreciated at 10% per year on the written-down-value method.

• Value after n years = 8,00,000 × (0.9)ⁿ. With 0.9⁵ = 0.59049 → ₹4,72,392 after 5 years.
• Total depreciation charged in 5 years = 8,00,000 − 4,72,392 = ₹3,27,608.
• Separately, the GP 3, 6, 12, … to 10 terms: S₁₀ = 3(2¹⁰ − 1)/(2 − 1) = 3 × 1023 = 3069.

Example 3 — Infinite GP: rebound distance and a recurring decimal (a) A ball dropped from 10 m rebounds to 60% of its previous height each time; find the total distance travelled before coming to rest.

• Distance = 10 + 2(10 × 0.6 + 10 × 0.36 + ⋯) = 10 + 2 × [6/(1 − 0.6)] = 10 + 2 × 15 = 40 m.

(b) Express 0.4̄5̄ as a fraction: it is 45/100 + 45/10000 + ⋯, a GP with a = 0.45, r = 0.01 → S = 0.45/0.99 = 5/11. (c) Infinite AGP 1 + 2/3 + 3/9 + 4/27 + ⋯ (a = 1, d = 1, r = 1/3): S = 1/(2/3) + (1/3)/(4/9) = 1.5 + 0.75 = 9/4.

Example 4 — Logarithms: evaluation, digit count and an equation Given log 2 = 0.3010, log 3 = 0.4771.

• log 72 = log(2³ × 3²) = 3(0.3010) + 2(0.4771) = 0.9030 + 0.9542 = 1.8572.
• Digits in 2¹⁰⁰: log(2¹⁰⁰) = 100 × 0.3010 = 30.10 → digits = ⌊30.10⌋ + 1 = 31.
• Digits in 5⁵⁰: log 5 = 1 − 0.3010 = 0.6990 → 50 × 0.6990 = 34.95 → 35 digits.
• Solve log₂x + log₂(x − 2) = 3 → log₂[x(x − 2)] = 3 → x² − 2x − 8 = 0 → x = 4 or −2; x = −2 is rejected (argument negative), so x = 4.
• log 0.0025 = log(2.5 × 10⁻³) = 0.3979 − 3 = 3̄.3979 (characteristic −3, mantissa 0.3979).

Example 5 — CAGR by logs, and a minimum by AM–GM A firm's revenue rises from ₹4 crore to ₹9 crore in 5 years.

• CAGR = (9/4)^1/5 − 1. log(9/4) = 0.9542 − 0.6021 = 0.3521; ÷5 = 0.07042; antilog 0.07042 ≈ 1.176 → CAGR ≈ 17.6% p.a.
• Minimum of x + 4/x for x > 0: by AM ≥ GM, x + 4/x ≥ 2√4 = 4, attained at x = 2.
• Average speed for equal distances at 60 and 40 km/h = HM = 2(60)(40)/100 = 48 km/h (the AM, 50, is wrong here).

Where it is used

• Banking and NBFC products: EMI schedules, recurring deposits, SIP maturity values and loan amortisation tables in HDFC, SBI, ICICI and Bajaj Finance systems are GP sums; interest formulas follow RBI norms on compounding frequency.
• Finance and accounting roles: WDV depreciation under Schedule II of the Companies Act, 2013, net present value, annuity factors, and CAGR in company annual reports and equity research.
• Software and analytics: binary search complexity O(log n) and merge-sort O(n log n), tree heights, hashing and bit-length calculations (digits/bits via logs), log-transformed features and log-loss in machine learning.
• Engineering measurement: decibel scales in acoustics and communication, pH in chemistry, Richter magnitude in seismology (relevant to IS 1893 seismic zones), semi-log plots for RC decay and reliability.
• Operations and manufacturing: production ramp-ups and learning curves, stage-wise yield losses (a GP), exponential cooling/decay, and output averaging over equal batches (harmonic mean).
• Recruitment tests: quantitative sections of TCS NQT, Infosys, Wipro NLTH, Capgemini, Accenture, CAT/XAT, GATE general aptitude, IBPS/SBI PO and SSC CGL regularly carry 2–4 marks of progressions and logarithms.

Interview questions

• Q: What is the difference between a sequence and a series? A: A sequence is an ordered list of terms; a series is the sum of the terms of a sequence.
• Q: When does an infinite GP have a finite sum? A: Only when the common ratio satisfies |r| < 1, and then S∞ = a/(1 − r); otherwise the terms do not shrink fast enough and the sum diverges.
• Q: Why can we not write a simple sum formula for an HP? A: The reciprocals form an AP, but the sum of reciprocals of an AP has no elementary closed form, so HP problems are solved term-wise through the reciprocal AP.
• Q: Prove or state the relation among A, G and H of two numbers. A: For positive a and b, A = (a+b)/2, G = √(ab), H = 2ab/(a+b); then G² = A·H and A ≥ G ≥ H, with equality only when a = b.
• Q: A car covers equal distances at 60 and 40 km/h. Average speed? A: 48 km/h — the harmonic mean, because equal *distances* mean unequal times, so the arithmetic mean 50 is wrong.
• Q: How do you find the number of digits in 3⁴⁰? A: ⌊40 × log₁₀3⌋ + 1 = ⌊40 × 0.4771⌋ + 1 = ⌊19.084⌋ + 1 = 20 digits.
• Q: Why is log of a negative number undefined? A: For a positive base, aˣ is always positive, so no real x gives a negative or zero result; such logs exist only in the complex domain.
• Q: What are characteristic and mantissa? A: The integral and the decimal parts of a common logarithm; the mantissa depends only on the digits of the number and the characteristic only on the position of the decimal point.
• Q: State the change-of-base formula and one use. A: log₍a₎N = log₍b₎N / log₍b₎a; it lets a calculator with only log and ln evaluate log₂N, as in computing bits needed to store a number.
• Q: If log₍1/3₎x > 2, what is x? A: The base is less than 1, so the inequality reverses: 0 < x < (1/3)² = 1/9.
• Q: Which progression models compound interest, and which models simple interest? A: Compound interest balances form a GP with ratio (1 + i); simple interest balances form an AP with common difference P·i.
• Q: The sum of three numbers in AP is 27 and their product 648 — find them. A: Take a−d, a, a+d: 3a = 27 → a = 9; 9(81 − d²) = 648 → d² = 9 → d = ±3, giving 6, 9, 12.

Common mistakes

• Using n instead of (n−1) in aₙ = a + (n−1)d, or dropping the "+1" in the count n = (l − a)/d + 1.
• Applying S∞ = a/(1 − r) when |r| ≥ 1, where no finite sum exists.
• Averaging speeds or rates with the arithmetic mean when the harmonic mean is required (equal distances, equal work quantities).
• Writing log(M + N) = log M + log N or log(M/N) = log M / log N — only products, quotients and powers have laws.
• Forgetting that the inequality direction reverses when the logarithm base lies between 0 and 1.
• Accepting every algebraic root of a log equation without checking that each argument stays positive and each base stays valid.
• Mishandling the bar notation: 3̄.3979 means −3 + 0.3979 = −2.6021, not −3.3979.

Related topics

Simple and compound interest · Algebra: linear and quadratic equations · Averages, mixtures and alligation · Number system and divisibility · Squares, cubes, roots and surds · Permutations and combinations