The first term of an A.P. is 5, the last term is 45 and the sum of all the terms is 400. Find the number of terms and the common difference of the A.P.
Generated by claude-sonnet-4-6 · 2026-06-15 10:31 · grounding rag
Model Answer
Given: $a = 5$, $l = 45$, $S = 400$
Finding n:
Using $S = \dfrac{n}{2}(a + l)$
$$400 = \frac{n}{2}(5 + 45)$$
$$400 = \frac{n}{2} \times 50 = 25n$$
$$n = 16$$
Finding common difference d:
Using $a_n = a + (n-1)d$
$$45 = 5 + (16-1)d$$
$$40 = 15d$$
$$d = \frac{40}{15} = \frac{8}{3}$$
∴ Number of terms = 16 and common difference $d = \dfrac{8}{3}$
Source: Arithmetic Progressions, Exercise 5.3 Q5
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Explanation
- Examiners expect two clear steps: first find $n$ using the sum formula with last term $S = \frac{n}{2}(a+l)$, then find $d$ using the $n$th term formula $a_n = a + (n-1)d$.
- Write the formula, substitute values, and solve — show all working for full 3 marks.
- A common mistake is using $S = \frac{n}{2}[2a+(n-1)d]$ first (which has two unknowns); always use $S = \frac{n}{2}(a+l)$ when the last term is given.