The sum of the first n terms of an arithmetic progression is given by Sn = 2n² + n. Find the first term and the common difference
To find the first term and common difference of the arithmetic progression, we need to use the formula for the sum of the first n terms of an arithmetic progression, which is:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the first n terms, a is the first term, and d is the common difference.
Given that Sn is given by Sn = 2n² + n, we can substitute this into the formula:
2n² + n = (n/2)(2a + (n-1)d)
Simplifying this equation:
2n² + n = n(2a + (n-1)d)/2
Multiplying both sides by 2 to eliminate the fraction:
4n² + 2n = n(2a + (n-1)d)
Expanding the right side:
4n² + 2n = 2an + nd - nd + d
Rearranging terms:
4n² + 2n = 2an + d(n-1) + d
Grouping common terms:
4n² + 2n = (2a + d)n + (d - a)
Comparing coefficients of n on both sides:
4 = 2a + d
Comparing constant terms on both sides:
0 = d - a
From the first equation, we can solve for a in terms of d:
2a = 4 - d
a = (4 - d)/2
a = 2 - d/2
Substituting this into the second equation:
0 = d - (2 - d/2)
0 = d - 2 + d/2
2 = d/2
4 = d
Therefore, the first term (a) is 2 - d/2 = 2 - 4/2 = 2 - 2 = 0, and the common difference (d) is 4.