evaluate the series ,
6 ‡” (2k+2)
k=1
To evaluate the given series, we can start by rewriting it in a more familiar form. The series can be expressed as:
6(2^1) + 6(2^3) + 6(2^5) + ...
Now, let's analyze the pattern:
1. The constant multiplier is 6 in each term.
2. The power of 2 in each term starts with 1 and increases by 2 in each subsequent term (1, 3, 5, ...).
To evaluate the series, we can calculate the individual terms and sum them up.
First, let's find the general formula for each term using the given pattern:
Term(n) = 6(2^(2n-1))
Now, we can substitute the values of k with n and evaluate the series.
The expression for the nth term is given as:
Term(n) = 6(2^(2n-1))
So, the series can be written as:
Sum(n) = Term(1) + Term(2) + ... + Term(n)
= 6(2^(2*1-1)) + 6(2^(2*2-1)) + ... + 6(2^(2n-1))
To find the sum of n terms, we can use the formula for the sum of a geometric series:
Sum(n) = a(r^n - 1) / (r - 1)
where a is the first term and r is the common ratio.
In this case, the first term (a) = 6 and the common ratio (r) = 4 since 2^(2n-1) = (2^2)^(n-1) = 4^(n-1).
Therefore, the sum of the series is:
Sum(n) = 6(4^n - 1) / (4 - 1)
= 2(4^n - 1)
So, the sum of the series can be represented as 2(4^n - 1), where n represents the number of terms in the series.