Respuesta :
Answer: The required expression is
[tex]S_{16}=\dfrac{5(4^{16}-1)}{3}.[/tex]
Step-by-step explanation: We are given to write an expression that gives the following sum :
the sum of first 16 terms of a geometric sequence with first term 5 and common ratio 4.
We know that
the sum of first n terms of a geometric sequence with first term a and common ratio r with |r| > 1 is given by
[tex]S_n=\dfrac{a(r^n-1)}{r-1}.[/tex]
For the given geometric sequence, we have
first term, a = 5 and common ratio, r = 4.
So, |r| = |4| = 4 > 1.
Therefore, the required expression for the sum of first 16 terms is given by
[tex]S_{16}=\dfrac{5(4^16)-1}{4-1}=\dfrac{5(4^{16}-1)}{3}[/tex]
Thus, the required expression is [tex]S_{16}=\dfrac{5(4^{16}-1)}{3}.[/tex]
