If in a GP, the sum of the first 18 terms is equal to the sum of first 22 terms and the sum of the first 19 terms is 65, then what will be the sum of first 4 terms? (Note: $$i = \sqrt{-1}$$)

Sum to n terms in GP is $$a\cdot\frac{\left(r^n-1\right)}{r-1}$$

The sum of the first 18 terms is equal to 22 terms.

So $$a\cdot\frac{\left(r^{18}-1\right)}{r-1} = a\cdot\frac{\left(r^{24}-1\right)}{r-1}$$

This will be reduced to $$\left(r^{18}-1\right)=\ r^{22}-1$$

Reducing this we get $$r^{4\ }=1$$

Now r has 4 posibilities 1, -1, i,-i.

1 is impossible as that would lead to a zero denominator in the sum of terms formulae.

In all the other cases of r, the sum of the first four values would be zero.