6∑6(2^K-1) K=1
To find the value of the expression 6∑6(2^K-1) K=1, we can start by understanding the notation used in the expression.
The ∑ symbol represents the summation notation, which indicates that we need to take the sum of the terms that follow. In this case, we are summing the terms 6(2^K-1) as K takes on values from 1 to 6.
Let's break down the expression step by step:
For K = 1:
Substituting K = 1 into the expression, we get:
6(2^1-1) = 6(2-1) = 6(1) = 6
For K = 2:
Substituting K = 2 into the expression, we get:
6(2^2-1) = 6(4-1) = 6(3) = 18
For K = 3:
Substituting K = 3 into the expression, we get:
6(2^3-1) = 6(8-1) = 6(7) = 42
And so on, until K = 6:
For K = 6:
Substituting K = 6 into the expression, we get:
6(2^6-1) = 6(64-1) = 6(63) = 378
To find the overall value of the summation, we need to add up all these individual results:
6 + 18 + 42 + ... + 378
We can use the formula for the sum of an arithmetic series to simplify this calculation. The formula is:
Sum = (n/2) * (first term + last term)
In this case, n = 6 (since there are 6 terms), the first term is 6, and the last term is 378.
Plugging in these values, we have:
Sum = (6/2) * (6 + 378) = 3 * 384 = 1152
Therefore, the value of the expression 6∑6(2^K-1) K=1 is 1152.