sum of first 5 terms and sum of first 7 terms in an AP

[pram]

In an AP, the sum to first 5 terms is equal to the sum to first 7 terms which is 167. In the same AP, the sum to first 10 terms is equal to 235. Find the sum to first 40 terms.

Can you give me a solution?

[Math Tutor]

In an AP, the sum to first 5 terms is equal to the sum to first 7 terms which is 167.

Is this true?
Or you would like to write that the sum of first 5 terms is equal to the 7-th term = 167?

[pram]

Or you would like to write that the sum of first 5 terms is equal to the 7-th term = 167?

No. original statement is correct. "sum of first 5 terms = sum of first 7 terms = 167"

[Math Tutor]

Why we have: the sum to first 10 terms is equal to 235 ?

[Math Tutor]

a1 + a2 + a3 + a4 + a5 = a1+ a2 + a3+ a4 + a5 + a6 + a7
so
a6 + a7 = 0
a6 = a1 + 5d
a7 = a1 + 6d

then a1 = 11d/2

from a1 + a2 + a3 + a4 + a5 = 167
we have a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) + (a1 + 4d) = 167
5a1 + 10d = 167
5*11d/2 + 10d = 167
65d = 167 * 2
d = 334/65
a1 = 1837/65

Here is the formula for sum of numbers

[tex]S = \frac{1}{2}(2a_1 + d(n-1))n[/tex]
n=39