Question

FIND The 6th And 5th Term Of The A.P Whose First Term Is 6 And Common Difference Is ?

Answer 1

You are expecting a miracle, aren't you?

Answer 2

sum of n terms of 1st = (n/2)(10 + 36(n-1))
= (n/2)(36n - 26)
sum of 2n terms of 2nd = (2n/2)(72 + 5(2n-1))
= n(10n + 67)

(n/2)(36n - 26) = n(10n + 67)
divide by n, and multiply by 2
36n - 26 = 20n + 134
16n = 160
n = 10

check:
sum(10 for 1st) = 5(10 + 36(9)) = 1670
sum(20 for 2nd) = 10(72 + 19(5)) = 1670
all is good!

Answer 3

Let the common difference be d.
The sum of the first four terms is
5 + (5+d) + (5+d+d) + (5+d+d+d)
= 20 + 6d
The sum of the next 4 terms is
5+4d + 5+5d + 5+6d + 5+7d = 20 + 22d

You have been told that
20 +6d = (1/2)(20 + 22d)= 10 + 11d
Therefore
5d = 10
d = 2

Check: does
5+7+9+11 = (1/2)(13+15+17+19) ?
32 = (1/2)(64)