Let a1 be the first term, am be the mth term, an be the nth term and d be the common difference between terms
The sum of the first m terms =
m^2 = m/2 * (a1 + am) → 2m = a1 + am (1)
The sum of the first n terms =
n^2 = n/2 * (a1 + an) → 2n = a1 + an (2)
Subtract (2) from (1)
2m - 2n = am - an (3)
And the mth term is given by
am = a1 + d (m - 1) → am = a1 + dm - d (4)
And the nth term is given by
an = a1 + d(n - 1) → an = a1 + dn - d (5)
Subtract (5) from (4)
am - an = dm - dn (6)
Set (3) and (6) equal to solve for d
2m - 2n = dm - dn
2(m - n) = d (m - n)
2 = d
Rearrange (1) and solve for a1
am = a1 + 2m - 2
2m - a1 = a1 + 2m - 2
2m = 2a1 + 2m - 2
0 = 2a1 - 2
2 = 2a1 divide by 2
1 = a1
Using (4) and (5) and subbing for a1 and d
am = 1 + 2m - 2 → am = 2m - 1
an = 1 + 2n - 2 → an = 2n - 1
So.....the ratio of the mth term to the nth term =
am : an = 2m - 1 : 2n - 1
