The nth term of a series is 2^n-1 +5n. Find the sum of the first n terms.
(Hint: This is an expression in terms of n).
well, the first n terms of 2^(n-1) is 2^n - 1
The first n terms of 5n is 5 * n(n+1)/2
So, the sum you want is
2^n - 1 + 5n(n+1)/2
To find the sum of the first n terms of a series, we need to find the sum of each term from 1 to n.
The nth term of the series is given by the expression 2^(n-1) + 5n.
Let's write out the first few terms of the series to see the pattern:
First term (n=1): 2^(1-1) + 5(1) = 2^0 + 5 = 1 + 5 = 6
Second term (n=2): 2^(2-1) + 5(2) = 2^1 + 10 = 2 + 10 = 12
Third term (n=3): 2^(3-1) + 5(3) = 2^2 + 15 = 4 + 15 = 19
From these examples, we can observe a pattern in the terms of the series:
First term: 6
Second term: 12 = 6 + 6
Third term: 19 = 12 + 7
We can see that each term of the series is obtained by adding a constant increase to the previous term. And this constant increase is itself a combination of two components: 2^n-1 and 5n.
To find the sum of the first n terms, we can use the formula for the sum of an arithmetic series:
S = (n/2)(a + L)
In this case, 'n' is the number of terms, 'a' is the first term, and 'L' is the last term.
To find the last term, we can plug in 'n' into the expression for the nth term:
L = 2^(n-1) + 5n
Now let's plug in the values into the sum formula:
S = (n/2)(a + L) = (n/2)(a + (2^(n-1) + 5n))
Since we are finding the sum of the first n terms, the first term is a = 6.
Plugging in this value, we get:
S = (n/2)(6 + (2^(n-1) + 5n))
And that's the expression for the sum of the first n terms of the given series!