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?
\(n = 38,\ S = 6973\)
Given: First term \(a = 17\), last term \(l = 350\), common difference \(d = 9\).
We need to find the number of terms \(n\) and their sum \(S_n\).
For an AP, the nth term is
\[a_n = a + (n - 1)d\]
Here the last term is \(l = a_n = 350\). So:
\[350 = 17 + (n - 1) \cdot 9\]
Subtract 17 from both sides:
\[350 - 17 = 9(n - 1)\]
\[333 = 9(n - 1)\]
Divide by 9:
\[n - 1 = \dfrac{333}{9} = 37\]
So,
\[n = 37 + 1 = 38\]
Number of terms: \(n = 38\).
Sum of first \(n\) terms of an AP:
\[S_n = \dfrac{n}{2}(a + l)\]
Substitute \(n = 38\), \(a = 17\), \(l = 350\):
\[S_{38} = \dfrac{38}{2}(17 + 350)\]
\[S_{38} = 19 \times 367\]
Compute the product:
\[19 \times 367 = 6973\]
Sum of all terms: \(S = 6973\).
Conclusion: The AP has 38 terms and their sum is 6973.