find the sum of the following series
a. 5, 9, 13, ..., 101
and
c. 83, 80, 77, ..., 5
thank you!!
a. common difference = 4 ... number of terms = 25
... sum = [(101 + 5) / 2] * 25
b. common difference = -3 ... number of terms = 27
To find the sum of a particular series, you can use the formula for the sum of an arithmetic series. The formula is given by:
Sn = n/2 * (a1 + an)
Where Sn is the sum of the series, n is the number of terms in the series, a1 is the first term, and an is the last term.
Let's calculate the sum of each series:
a. Series: 5, 9, 13, ..., 101
Here, we can see that the first term, a1, is 5 and the last term, an, is 101. To find the number of terms, we need to determine the common difference between the terms. In this series, the common difference is 4 (9 - 5 = 4, 13 - 9 = 4, and so on).
To find the number of terms, we can use the formula:
n = (an - a1) / d + 1
Where d is the common difference.
Using this formula, we can find:
n = (101 - 5) / 4 + 1 = 24
Now, we have n = 24, a1 = 5, and an = 101. Plugging these values into the sum formula:
Sn = 24/2 * (5 + 101) = 12 * 106 = 1272
Therefore, the sum of the series is 1272.
c. Series: 83, 80, 77, ..., 5
Similar to the previous series, we need to find the first term, last term, and common difference.
In this series, the first term, a1, is 83, the last term, an, is 5, and the common difference is -3 (80 - 83 = -3, 77 - 80 = -3, and so on).
To find the number of terms, we use the formula:
n = (an - a1) / d + 1
Plugging in the values:
n = (5 - 83) / -3 + 1 = 27
Now, we have n = 27, a1 = 83, and an = 5. Plugging these values into the sum formula:
Sn = 27/2 * (83 + 5) = 13.5 * 88 = 1188
Therefore, the sum of the series is 1188.
Please note that in this example, I have assumed that the series is an arithmetic series. If you have more complex series, you may need to use different formulas or methods to find the sum.