Sequences and series

NUST NET Maths · Maths · Sequences, series and the binomial theorem. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

In an arithmetic sequence you add the same number d each time. In a geometric sequence you multiply by the same number r each time. Almost every question gives you two facts and expects you to find the first term a and then d or r.

A geometric series can be summed for ever only when its terms shrink, that is when |r| < 1. If |r| ≥ 1 there is no sum to infinity.

Means are terms placed between two numbers so that the whole list forms a sequence. A harmonic sequence is one whose reciprocals are in arithmetic sequence, so harmonic questions are solved by turning every term upside down first.

Rules to remember

Arithmetic: T_n = a + (n − 1)d; S_n = (n/2)[2a + (n − 1)d] = (n/2)(a + l), where l is the last term
Geometric: T_n = ar^{n − 1}; S_n = a(1 − rⁿ)/(1 − r), r ≠ 1
Sum to infinity: S_∞ = a/(1 − r), only for |r| < 1
From a sum formula: T_n = S_n − S_{n − 1}
Between a and b: arithmetic mean A = (a + b)/2; geometric mean G = ±√(ab); harmonic mean H = 2ab/(a + b)
G² = A × H; for two different positive numbers A > G > H, taking G as the positive root
In a geometric sequence, T_m ÷ T_n = r^{m − n}; in an arithmetic sequence, T_m − T_n = (m − n)d

Common mistake

Using n in place of n − 1 in the term formulas. The 10th term of an arithmetic sequence is a + 9d, and the 6th term of a geometric sequence is ar⁵.

Worked example

The 4th term of an arithmetic sequence is 11 and its 10th term is 29. The sum of its first 10 terms is

T₁₀ − T₄ = 6d, so 6d = 29 − 11 = 18 and d = 3.
T₄ = a + 3d = 11, so a = 11 − 9 = 2.
S₁₀ = (10/2)(a + l) = 5 × (2 + 29) = 5 × 31 = 155.

Answer: 155

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Arithmetic & Geometric Progressions

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