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.