Determine the sum of the arithmetic series 2+14+26+...+146
Question
number of terms: 1 + (146-2)/12 = 13
S13 = 13/2(2+146) = _____
To find the sum of an arithmetic series, we can use the formula:
Sum = (n/2)(2a + (n-1)d)
Where:
- Sum is the sum of the series
- n is the number of terms in the series
- a is the first term
- d is the common difference between the terms
In this case, we can see that the first term (a) is 2, and the common difference (d) is 12 (since each term is obtained by adding 12 to the previous term).
To find the number of terms (n), we can use the formula:
n = (Last Term - First Term) / Common Difference + 1
In this case, the last term is 146. Plugging in the values, we get:
n = (146 - 2) / 12 + 1 = 13.
Now we can substitute the values in the sum formula:
Sum = (13/2)(2(2) + (13-1)12)
= (13/2)(4 + 12*12)
= (13/2)(4 + 144)
= (13/2)(148)
= 13 * 74
= 962.
Therefore, the sum of the arithmetic series 2+14+26+...+146 is 962.
To find the sum of an arithmetic series, you need to know the first term (a), the last term (l), and the number of terms (n).
In this case, the first term is 2, and the common difference between consecutive terms is 12. We need to find the last term.
To find the last term (l), we can use the formula for the nth term in an arithmetic sequence: l = a + (n-1)d, where a is the first term, n is the number of terms, and d is the common difference.
In this case, a = 2 and d = 12. Let's find n by substituting the values into the formula and solving for n:
l = 2 + (n-1) * 12
l = 2 + 12n - 12
l = 12n - 10
Now, we need to find n when l = 146:
146 = 12n - 10
12n = 156
n = 13
So, the last term is l = 146, and the number of terms is n = 13.
Now, we can calculate the sum of the arithmetic series using the formula: S = (n/2)(a + l), where S is the sum, n is the number of terms, a is the first term, and l is the last term.
Let's substitute the values and calculate the sum:
S = (13/2)(2 + 146)
S = 6.5(148)
S = 962
Therefore, the sum of the arithmetic series 2+14+26+...+146 is 962.