In an AP, if \(S_n=3n^2+5n\) and \(a_k=164\), find \(k\).
\(k=27\)
Step 1: We know that \(S_n\) is the sum of the first \(n\) terms of the AP.
Step 2: The formula to find the \(n^{th}\) term is:
\(a_n = S_n - S_{n-1}\)
This is because the difference between the sum of first \(n\) terms and the sum of first \((n-1)\) terms gives the \(n^{th}\) term.
Step 3: Write \(S_n = 3n^2 + 5n\).
Now find \(S_{n-1}\):
\(S_{n-1} = 3(n-1)^2 + 5(n-1)\)
\(= 3(n^2 - 2n + 1) + 5n - 5\)
\(= 3n^2 - 6n + 3 + 5n - 5\)
\(= 3n^2 - n - 2\)
Step 4: Now find \(a_n = S_n - S_{n-1}\):
\(a_n = (3n^2 + 5n) - (3n^2 - n - 2)\)
\(= 3n^2 + 5n - 3n^2 + n + 2\)
\(= 6n + 2\)
Step 5: The general term is \(a_n = 6n + 2\).
We are told \(a_k = 164\).
So, \(6k + 2 = 164\).
Step 6: Solve for \(k\):
\(6k = 164 - 2 = 162\)
\(k = 162 / 6 = 27\)
Final Answer: \(k = 27\).