Answer: The required sum is 39991.
Step-by-step explanation: We are given to find the sum of the following 7-term geometric sequence:
1, -6, 36, . . ..
Here, the first term , a = 1
and
common ratio, 'r' is given by
[tex]d=\dfrac{-6}{1}=\dfrac{36}{-6}=~.~.~.=-6.[/tex]
We know that
the sum of first 'n' terms of a geometric sequence is given by
[tex]S_n=\dfrac{a(r^n-1)}{r-1},~r>1~~~~~~~~~\textup{or}~~~~~~~~~~~S_n=\dfrac{a(1-r^n)}{1-r},~r<1.[/tex]
For the given geometric sequence, r = -6 < 1.
Therefore, the sum of first 7 terms will be
[tex]S_7\\\\\\=\dfrac{a(1-r^n)}{1-r}\\\\\\=\dfrac{1(1-(-6)^7)}{1-(-6)}\\\\\\=\dfrac{1+279936}{1+6}\\\\\\=\dfrac{279937}{7}\\\\\\=39991.[/tex]
Thus, the required sum is 39991.
