TMUA 2020 Paper 2 Q14 — Worked Solution

P2·Sequences & Series, Logic & Quantifiers·7.0Official · 2020 · Q14

arithmetic-seriesnecessary-sufficientcounterexample

An arithmetic sequence

T

has first term

a

and common difference

d

, where

a

and

d

are non-zero integers.Property

P

is: for some positive integer $m$ , the sum of the first $m$ terms of the sequence is equal to the sum of the first $2 m$ terms of the sequence.For example, when

a = 11

and

d = - 2

, the sequence

T

has property

P

, because

11 + 9 + 7 + 5 = 11 + 9 + 7 + 5 + 3 + 1 + (- 1) + (- 3),

i.e. the sum of the first

4

terms equals the sum of the first

8

terms.Which of the following statements is/are true?

For

T

to have property

P

, it is sufficient that

a d < 0

For

T

to have property

P

, it is necessary that

d

is even.

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TMUA 2020 Paper 2 Question 14

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