Sequences

4 marks · Hard

Sophie decides to save money for a holiday by following a plan: she saves $15 in the first week and then increases how much she saves by $4 each week. She needs to save at least $350. Explain how she can work out the minimum number of weeks needed to reach her goal. Show your reasoning clearly.

Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.

Show answer

She sets up an inequality using the arithmetic series sum: , then solves for n. The minimum n that meets or exceeds 350 is the answer.

Must show: States correct arithmetic sum formula, Forms the correct inequality for n, Solves for n correctly to get minimum integer value, Explains the reasoning in context

Also accept: Alternative inequality setup and clear solution

Common mistakes: Using n instead of n-1 in the sum formula; Working with an approximate n and not finding the next whole number

Show mark scheme and worked solution

Marking points

M1 · 1 mark · Identifies the appropriate sum formula for arithmetic series involving n weeks.

M2 · 1 mark · Forms correct inequality: or equivalent.

M3 · 1 mark · Solves inequality to find n (with working shown, e.g., quadratic solved). (follow-through)

B1 · 1 mark · Arrives at minimum integer n that meets or exceeds the target; explains logic in context. (follow-through)

Worked solution

Her weekly savings form an arithmetic sequence with first term and difference .

Sum after n weeks: .

Set :

Solve: .

So she needs a minimum of 11 weeks to meet her goal.

She should plan for 11 weeks.