if the sum of the first n terms of an ap is S sem is equal to 3 and square + 5 and find its common difference

Answer: The common difference of the AP is 66.

Step-by-step solution

Step 1: Find the first term a1a_1

The sum of the first 11 term of an AP is equal to its first term, so a1=S1a_1 = S_1. Substituting n=1n = 1 into the given formula Sn=3n2+5nS_n = 3n^2 + 5n, we obtain 3(1)2+5(1)=83(1)^2 + 5(1) = 8.

Step 2: Find the sum of the first two terms S2S_2

To find the sum of the first 22 terms, we substitute n=2n = 2 into the given formula Sn=3n2+5nS_n = 3n^2 + 5n. This gives S2=3(4)+10=22S_2 = 3(4) + 10 = 22.

Step 3: Find the second term a2a_2

Since the sum of the first 22 terms is a1+a2=S2a_1 + a_2 = S_2, the second term is given by a2=S2−S1a_2 = S_2 - S_1. Subtracting 88 from 2222 gives a2=14a_2 = 14.

Step 4: Calculate the common difference dd

The common difference dd of an AP is the difference between any two consecutive terms, d=a2−a1d = a_2 - a_1. Therefore, d=14−8=6d = 14 - 8 = 6.

Solve your own maths question free →