Find the sum of a finite geometric sequence from n = 1 to n = 5, using the expression -3(4)n - 1.
Solution:
In a Geometric Sequence, each term is found by multiplying the previous term by a constant. a, ar, ar2, ar3, ...
Given geometric expression -3(4)n - 1 from n = 1 to 5.
To find the sum of geometric expression from n = 1 to n = 5 is
S\(_n\) = \(\sum_{n=1}^{5} -3(4)^n - 1\)
S\(_n\) = - 3(4)1 - 1 - 3(4)2 - 1 - 3(4)3 - 1 - 3(4)4 - 1 - 3(4)5 - 1.
S\(_n\) = - 5 - 3(4 + 42+ 43+ 43+ 45 )
Consider the the geometric progression inside the brackets. Here a = 4 and n =5, r = 4
sum of gp if r >1 =\(\dfrac{a(r)n-1}{r-1}\),
S\(_n\)= - 5 - 3(4(4)5 -1]/ 3)
S\(_n\)= - 5 - 3(1364)
S\(_n\)=-4097
Find the sum of a finite geometric sequence from n = 1 to n = 5, using the expression - 3(4)n - 1.
Summary:
The sum of a finite geometric sequence from n = 1 to n = 5, using the expression - 3(4)n - 1 is -4097.