The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
Step-by-Step Solution
Key Concept: Use the nth term formula $a_n = a + (n-1)d$ to find the number of terms $n$, and then use the sum formula $S_n = \frac{n}{2}(a + l)$ where $l$ is the last term.
1. Given data\
First term $a = 17$, last term $l = 350$, common difference $d = 9$.\
2. Find the number of terms $n$ using the nth term formula:\
$$\begin{aligned}
l &= a + (n-1)d \\
350 &= 17 + (n-1)\times 9 \\
350 - 17 &= 9(n-1) \\
333 &= 9(n-1) \\
n-1 &= \frac{333}{9} = 37 \\
n &= 37 + 1 = 38
\end{aligned}$$\
Hence, the AP contains 38 terms.\
3. Find the sum of the AP using the sum formula:\
$$\begin{aligned}
S_n &= \frac{n}{2}\,(a + l) \\
S_{38} &= \frac{38}{2}\,(17 + 350) \\
&= 19 \times 367 \\
&= 6973
\end{aligned}$$\
Therefore, the sum of all the terms is 6973.
Correct Answer: Number of terms $n = 38$, Sum $S_{38} = 6973$