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A sequence is defined by $u_1 = a$ and $u_{n+1} = -1 \times u_n$ Find $$ \sum_{n=1}^{95} u_n $$ Circle your answer - AQA - A-Level Maths Mechanics - Question 3 - 2021 - Paper 2

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A-sequence-is-defined-by--$u_1-=-a$-and-$u_{n+1}-=--1-\times-u_n$--Find---$$-\sum_{n=1}^{95}-u_n-$$--Circle-your-answer-AQA-A-Level Maths Mechanics-Question 3-2021-Paper 2.png

A sequence is defined by $u_1 = a$ and $u_{n+1} = -1 \times u_n$ Find $$ \sum_{n=1}^{95} u_n $$ Circle your answer. −a 0 a 95a

Worked Solution & Example Answer:A sequence is defined by $u_1 = a$ and $u_{n+1} = -1 \times u_n$ Find $$ \sum_{n=1}^{95} u_n $$ Circle your answer - AQA - A-Level Maths Mechanics - Question 3 - 2021 - Paper 2

Step 1

Find the sequence terms

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Answer

Given the sequence defined by ( u_1 = a ) and ( u_{n+1} = -1 \times u_n ), we can establish the first few terms:

  • ( u_1 = a )
  • ( u_2 = -a )
  • ( u_3 = a )
  • ( u_4 = -a )

This indicates the sequence alternates between ( a ) and ( -a ).

Step 2

Determine the sum of the first 95 terms

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Answer

The sequence has alternating terms, so we can group them:

  • In the first 95 terms, there are 48 pairs of ( a ) and ( -a ) and one additional term which will be ( a ). Thus the calculation is:

∑n=195un=(48×(a+(−a)))+a=0+a=a\sum_{n=1}^{95} u_n = (48 \times (a + (-a))) + a = 0 + a = a

Step 3

Final answer

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Answer

The final answer is ( a ).

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