Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 6

Table of contents
- a 6
  - Solution a
- b 6
  - Solution b

a

Solution

Group consecutive terms in pairs. The

j

th pair, for

j = 1, \dots, 25

, is

- \frac{4 j - 3}{2} + \frac{4 j - 1}{2} = 1.

There are

25

such pairs in this finite sum, so the answer is

25

.

b

Solution

This geometric series has first term

1/27

and common ratio

1/3

, whose absolute value is below one. Consequently its sum is

\frac{1/27}{1 - 1/3} = \frac{1}{18} .

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