Sequences & Series

8 methods · 20 questions

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Many TMUA sums are a geometric series wearing a costume (a trigonometric sign factor, a recurring positional digit, a derivative operator, a parameter). Strip the costume, identify the constant ratio $r$ and first term $a$ , confirm $∣ r ∣ < 1$ , and finish with $S_{\infty} = \frac{a}{1 - r}$ or $S_{N} = \frac{a ( 1 - r ^{N} )}{1 - r}$ .

Trigger: the summand carries a hidden constant multiplier from one index to the next.

Instances:
(i) a $(- 1)^{n}$ or $cos (nπ)$ factor gives a geometric series with negative ratio;
(ii) a recurring base- $b$ expansion is geometric with ratio $b^{- (period)}$ , recovered by multiplying by $b^{period}$ and subtracting;
(iii) a derivative or scaling recurrence produces successive powers, i.e. a constant ratio.

Linked questions (3)

TMUA 2016 Paper 1 · Q14

The terms of an infinite series S are formed by adding together the corresponding terms in two infinite geometric series, T and U. The first term of T and the first term of U are each 4. In order, the first three terms of the combined series S are

8

3

, and

\frac{5}{4}

. What is the sum to infinity of S?

A $\frac{32}{5}$
B $\frac{20}{3}$
C $\frac{64}{5}$
D $\frac{40}{3}$
E $16$
F $32$

TMUA 2016 Paper 2 · Q6

The sequence of functions

f_{1} (x), f_{2} (x), f_{3} (x), \dots

is defined as follows:

f_{1} (x) = x^{10}

f_{n + 1} (x) = x f_{n}^{'} (x) for n \geq 1

where

f_{n}^{'} (x) = \frac{d f _{n} ( x )}{d x}

.

Find the value of

n = 1 \sum 20 f_{n} (x)

A $\frac{x ^{10} ( x ^{20} - 1 )}{x - 1}$
B $\frac{x ^{10} ( x ^{21} - 1 )}{x - 1}$
C $(\frac{1 0 ^{20} - 1}{9}) x^{10}$
D $(\frac{1 0 ^{21} - 1}{9}) x^{10}$
E $(\frac{( 10 x ) ^{20} - 1}{10 x - 1}) x^{10}$
F $(\frac{( 10 x ) ^{21} - 1}{10 x - 1}) x^{10}$
G $x^{10} + x^{9} + x^{8} + \dots + x + 1$
H $x^{10} + 10 x^{9} + (10 \times 9) x^{8} + \dots + (10 \times 9 \times \dots \times 2) x + (10 \times 9 \times \dots \times 2 \times 1)$

TMUA 2017 Paper 2 · Q3

The first term of a geometric progression is

23

and the fourth term is

\frac{9}{4}

. What is the sum to infinity of this geometric progression?

A $- 2 (2 - 3)$
B $4 (23 - 3)$
C $\frac{16 ( 8 3 + 9 )}{37}$
D $\frac{4 ( 2 3 - 3 )}{7}$
E $\frac{4 ( 2 3 + 3 )}{7}$
F $2 (2 + 3)$
G $4 (23 + 3)$