Question

The first term of an AP is $5,$ the last term is $45$ and the sum is $400.$ Find the number of terms and the common difference.

Answer 1

Step 1: Find the number of terms in the series.

first term, $a=5$

last term, $l=45$

Sum of the AP, $S_n=400$

As we know, the sum of the AP formula is;

$$\begin{gathered} S_n=\frac{n}{2}(a+l) \\ 400=\frac{n}{2}(5+45) \\ 400=\frac{n}{2}\left(50\right) \\ \therefore n=16 \end{gathered}$$

Number of terms, $n=16$

Step 2: Find the common difference of the series.

As we know, the last term of the AP series can be written as; $l=a+(n-1)d$

$$\begin{gathered} 45=5+(16-1)d \\ 40=15d \\ \therefore d=\frac{40}{15} \\ =\frac{8}{3} \end{gathered}$$

Hence, the number of terms in the series is $16,$and the common difference is $\frac{8}{3}.$