question_answer

The first term of an A.P. is 5, the last term is 45 and the sum of all its terms is 400. Find the number of terms and the common difference of the A.P.

Answer:

Given, \[a=5,{{a}_{n}}=45,{{S}_{n}}=400\]

We have,           \[{{S}_{n}}=\frac{n}{2}[a+{{a}_{n}}]\]

\[\Rightarrow \]               \[400=\frac{n}{2}[5+45]\]

\[\Rightarrow \]               \[400=\frac{n}{2}[50]\]

\[\Rightarrow \]               \[25n=400\Rightarrow n=\frac{400}{25}\]

\[\Rightarrow \]                   \[n=16\]

Now,                 \[{{a}_{n}}=a+(n-1)d\]

\[\Rightarrow \]               \[45=5+(16-1)d\]

\[\Rightarrow \]           \[45-5=15\,d\]

\[\Rightarrow \]              \[15\,d=40\]

\[\Rightarrow \]                 \[d=\frac{8}{3}\]

So,                   \[n=16\] and \[d=\frac{8}{3}\]