The sum of the first n terms of an arithmetic progression is given by S_n = 3n^2 - n. Find the 5th term of the sequence.
Correct: A
We are given the sum of the first n terms of an arithmetic progression as S_n = 3n^2 - n.
The nth term of an arithmetic progression, a_n, can be found using the formula a_n = S_n - S_{n-1} for n > 1.
For the first term, a_1 = S_1.
Let's find the first term, a_1:
a_1 = S_1 = 3(1)^2 - 1 = 3 - 1 = 2.
To find the 5th term, a_5, we use:
a_5 = S_5 - S_4.
First, calculate S_5:
S_5 = 3(5)^2 - 5 = 3(25) - 5 = 75 - 5 = 70.
Next, calculate S_4:
S_4 = 3(4)^2 - 4 = 3(16) - 4 = 48 - 4 = 44.
Now, calculate a_5:
a_5 = S_5 - S_4 = 70 - 44 = 26.
Alternatively, we could find the common difference d.
a_2 = S_2 - S_1.
S_2 = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10.
a_2 = 10 - 2 = 8.
The common difference d = a_2 - a_1 = 8 - 2 = 6.
Now, use the formula for the nth term of an arithmetic progression: a_n = a_1 + (n-1)d.
a_5 = a_1 + (5-1)d = a_1 + 4d.
a_5 = 2 + 4(6) = 2 + 24 = 26.
Both methods yield the same result.