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An arithmetic sequence has first term a and common difference d - AQA - A-Level Maths: Pure - Question 9 - 2018 - Paper 1
Question 9
An arithmetic sequence has first term a and common difference d. The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 term... show full transcript
Worked Solution & Example Answer:An arithmetic sequence has first term a and common difference d - AQA - A-Level Maths: Pure - Question 9 - 2018 - Paper 1
Step 1
Given that the sixth term of the sequence is 25, find the smallest possible value of a.
Answer
To find the sixth term of the sequence, we use the expression for the nth term:
For the sixth term, where :
We know from the problem statement that :
Rearranging this gives:
Now, substituting this expression for into the quadratic equation we derived in part (a):
Expanding and simplifying gives:
- Substitute into .
- Minimize the obtained expression for .
Eventually, setting will yield:
Verifying with the quadratic equation shows that this is the smallest achievable value for . Thus, the solution yields: \smallest possible value of a = -55.