Question

the sum of first 13 terms of a.p. is 21 and the sum of first 21 terms is 13 . find the sum of the first 34 terms.

Answer 1

The sum of the first 13 terms is 13a + 78d = 21. 
The sum of the first 21 terms is 21a + 210d = 13 
Multiply the first equation by 21 and the second by 13. 
273a + 1638d = 441 
273a + 2730d = 169 
Subtract. 
1092d = -272 
d = -68/273 
Substitute into the first equation above. 
13a -136/7 = 21 
a = 283/91 
So the nth sum is 283n/91 - 34n(n-1)/273. 

Hence the sum of the first 34 terms is: 
283*34/91 - 34*34*33/273 
= 9622/91 - 12716/91 
= -3094/91 
= -34