In an AP, if \(S_n=n(4n+1)\), find the AP.
AP: 5, 13, 21, … (\(a=5, d=8\))
Step 1: Recall the meaning of \(S_n\)
\(S_n\) represents the sum of the first \(n\) terms of the AP.
Step 2: Find the first term
Put \(n = 1\) in the given formula:
\(S_1 = 1(4(1) + 1) = 1(4 + 1) = 5\).
So, the first term \(a = 5\).
Step 3: Find the second term
Put \(n = 2\):
\(S_2 = 2(4(2) + 1) = 2(8 + 1) = 2 \times 9 = 18\).
But \(S_2\) means the sum of the first two terms:
\(S_2 = a_1 + a_2 = 5 + a_2 = 18\).
So, \(a_2 = 18 - 5 = 13\).
Step 4: Find the common difference
The common difference \(d = a_2 - a_1 = 13 - 5 = 8\).
Step 5: Write the AP
The AP is: \(5, 13, 21, …\).