An Arithmetic Progression Savings Problem

The signal for an arithmetic progression in a word problem is a fixed amount added each period, here, a saving that grows by the same sum every week. Read off the first term $a$ and common difference $d$, then apply the sum formula.

A person saves money each week for $20$ weeks. They save RM$50$ in the first week, and each week after that they save RM$10$ more than the week before. Find the total amount saved over the $20$ weeks. [4]

The working

Step 1, identify the AP. A fixed RM$10$ more each week means a common difference: $a = 50, \qquad d = 10, \qquad n = 20 \quad \text{(M1)}$

Step 2, apply the sum formula $S_n = \dfrac{n}{2}\big[2a + (n - 1)d\big]$: $S_{20} = \frac{20}{2}\big[2(50) + 19(10)\big] \quad \text{(M1)}$

Step 3, evaluate: $= 10\big[100 + 190\big] = 10 \times 290 = 2900 \quad \text{(A1, A1)}$

So the total saved is RM2900.

Where the marks are won and lost

• Recognise the AP and read off $a = 50$, $d = 10$ correctly. “RM10 more each week” is the common difference, not the first term.
• Use $(n - 1)d = 19 \times 10$, not $20 \times 10$. The off-by-one on $n - 1$ is the classic slip.
• The total is a sum ($S_{20}$), not the 20th term, read the question: “total amount saved”.

Common mistakes

• Computing the 20th term instead of the sum.
• Using $20 \times 10$ instead of $19 \times 10$ inside the bracket.
• Treating “RM10 more each week” as a percentage (that would be a GP).

Full method: Arithmetic Progressions notes. Topic home: Series pillar.