Question

the 6 term of an ap is zero. prove that its 31 first term is 5 times the 11 term

Answer 1

let the first term is a and 6th term is a+5d=0 that means a=-5d now sinply substitute the values we get condition satisfied

Answer 2

Let's take an AP with first term a, and common difference d. So, 6th term will be a+5d which equals 0. So, a+5d=0 which means a=-5d.
So, nth term is a+(n-1)d = - 5d + nd - d = (n-6)d.
Now, 31st term is (31-6)d = 25d = 5*(5d).

5*(5d) = 5*(11-6)d
But (11-6)d is the 11th term. Thus, a31 = 5*a11.