Sequences and series

NUST NET (Engineering) · Maths · Sequences, series and the binomial theorem. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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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

Practice questions

Try each one, then open the answer.

1. Three real geometric means are inserted between 1/3 and 27. Their product is

A
9
B
27
C
81
D
729

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Answer: B. 1/3, G₁, G₂, G₃, 27 are in G.P. with r⁴ = 81, so the means are 1, 3, 9 (or −1, 3, −9). Either way the product is 27; 9 is only the middle mean.

2. An infinite geometric series has sum 4, and the series formed by the squares of its terms has sum 16/3. The first term is

A
1
B
2
C
3
D
4/3

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Answer: B. a/(1 − r) = 4 and a²/(1 − r²) = 16/3. Dividing the second by the first gives a/(1 + r) = 4/3; with a = 4(1 − r) this gives r = 1/2 and a = 2.

3. Two positive numbers have arithmetic mean 13 and geometric mean 12. The numbers are

A
9 and 16
B
6 and 20
C
12 and 14
D
8 and 18

Show answer

Answer: D. a + b = 26 and ab = 144, so a and b are the roots of t² − 26t + 144 = 0, i.e. (t − 8)(t − 18) = 0. 9 and 16 have product 144 but sum 25.

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Arithmetic & Geometric Progressions

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