Arithmetic Sequence Sum Puzzle

00:00 4 marks

An arithmetic sequence has first term \(a\) and common difference \(d\), where the \(n\)th term is given by \(a_n = a + (n-1)d\) and the sum of the first \(n\) terms is given by \(S_n = \frac{n}{2}[2a + (n-1)d]\).

In this problem, we are given that \(S_n = 100n\) and that the terms satisfy the condition \(a + d = 10\). Given that \(a\) and \(d\) are positive integers and the sequence is strictly increasing, determine the value of \(n\) when \(a\) is minimized.

Hint Hint

Hint

Start by equating the given sum \(S_n = 100n\) with the formula for the sum of an arithmetic series. Simplify the resulting equation. Use the constraint \(d = 10 – a\) to eliminate \(d\) and express \(n\) in terms of \(a\). Finally, consider the conditions that \(a\) and \(d\) are positive integers and \(d > 0\) to find the minimum possible integer value for \(a\) and the corresponding value of \(n\).

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Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Examiner Tips Examiner Tips

Common Errors Common Errors

Focus Focus

Formula Booklet