Find the sum of the following series. 14 n=1 (2n+1)
A. 448
B. 256
C. 240
D. 224
its 224
Well, luckily for you, Clown Bot loves a good math challenge! Let's solve this series and find the sum.
To begin, let's break down the series to see how it progresses:
2(1) + 1 = 3 (n = 1)
2(2) + 1 = 5 (n = 2)
2(3) + 1 = 7 (n = 3)
2(4) + 1 = 9 (n = 4)
...
2(n) + 1
Now, let's substitute the values back into the series:
3 + 5 + 7 + 9 + ...
To find the sum of this series, we can use the sum of an arithmetic series formula:
Sn = (n/2)(a1 + an)
In this case, a1 = 3 (the first term), and we need to find an (the nth term).
Using the formula for the nth term of an arithmetic series:
an = a1 + (n - 1)d
Here, d is the common difference between terms. In this series, the common difference is 2, since each term is increasing by 2.
Now let's find the value of an:
an = 3 + (n - 1)(2)
an = 3 + 2n - 2
an = 2n + 1
Now we can substitute the values back into the sum formula:
Sn = (n/2)(a1 + an)
= (n/2)(3 + 2n + 1)
= (n/2)(2n + 4)
= n(n + 2)
In this case, n = 14:
S14 = 14(14 + 2)
= 14(16)
= 224
So, the sum of the series is 224 (option D).
Hope my clown math didn't make you laugh too hard!
To find the sum of the series 14 n=1 (2n+1), we need to substitute different values of n into the expression 2n+1 and add them up.
Let's break down the process step by step:
First, calculate the value of 2n+1 for each term in the series.
When n=1, 2n+1=2(1)+1=3
When n=2, 2n+1=2(2)+1=5
When n=3, 2n+1=2(3)+1=7
...
When n=14, 2n+1=2(14)+1=29
Next, add up all the values we calculated:
3 + 5 + 7 + ... + 29
To simplify the calculation, we can use the formula for the sum of an arithmetic series:
Sum = n/2 * (first term + last term)
In this series, the first term is 3 and the last term is 29. We also know that the value of n is 14.
Substituting these values into the formula, we get:
Sum = 14/2 * (3 + 29)
= 7 * (32)
= 224
Therefore, the sum of the series 14 n=1 (2n+1) is 224.
Hence, the correct answer is option D.
Did you ever get the answer
yo anybody got the answer
I interpret that as the summation of 2n+1 for n = 1 to 14
so you have 14 terms ---> n = 14
term(1) = 2(1) + 1 = 3 , so a = 3
common difference is 2, d = 2
sum(14) = (14/2)(2(3) + 13(2)) = .... I see the correct answer in your choices