Evaluate the following using the finite geometric sum formula.
7 E k=1 4^(k-1)
To evaluate the given expression using the finite geometric sum formula, we need to identify the common ratio (r), the first term (a), and the number of terms (n).
In this case, the common ratio (r) is 4, as each term is obtained by multiplying the previous term by 4.
The first term (a) can be found by substituting k = 1 into the given expression:
a = 4^(k-1) = 4^(1-1) = 4^0 = 1
Now, we need to determine the number of terms (n). In this case, the expression 7 E k=1 indicates that we sum up to k = 7. Therefore, n = 7 - 1 + 1 = 7.
Now, we can use the finite geometric sum formula:
S = a(r^n - 1) / (r - 1)
Substituting in the values we found:
S = 1(4^7 - 1) / (4 - 1)
S = (4^7 - 1) / 3
Calculating this expression:
S = (16384 - 1) / 3
S = 16383 / 3
S = 5461
Therefore, the value of the given expression is 5461.