Evaluate the following using the finite geometric sum formula.
7 E k=1 4^k-1
To evaluate the expression using the finite geometric sum formula, we can rewrite it as follows:
7 ∑ k=1 4^k-1
The finite geometric sum formula is given by:
S = a(1 - r^n) / (1 - r)
where S is the sum of the finite geometric series, a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 4^0 = 1 (since k starts from 1, we use the exponent of 0 to represent the first term), r = 4 (since each term is obtained by multiplying the previous term by 4), and n = 7 (since we have 7 terms in the series).
Using the formula, we can calculate the value as follows:
S = 1(1 - 4^7) / (1 - 4)
= 1(1 - 16384) / (1 - 4)
= 1(-16383) / (-3)
= 16383 / 3
= 5461
Therefore, the value of the given expression is 5461.