Question

Calculate the sum in the first 8 terms of the arithmetic sequence in which a_8=3 and the common difference is d=-5

Answer 1

To find the sum of the first 8 terms in an arithmetic sequence, we can use the formula for the sum of an arithmetic series:

S_n = (n/2)(2a_1 + (n-1)d),

where S_n is the sum of the first n terms, a_1 is the first term, and d is the common difference.

In this case, a_8 = 3 and d = -5. We are looking for the sum of the first 8 terms, so n = 8.

Plugging in the given values, we have:

S_8 = (8/2)(2a_1 + (8-1)(-5))
= 4(2a_1 + 7(-5))
= 4(2a_1 - 35).

We can solve for a_1 by substituting n = 8 and d = -5 into the formula a_n = a_1 + (n-1)d, where n is the term number:

a_8 = a_1 + (8-1)(-5)
3 = a_1 + 7(-5)
3 = a_1 - 35
a_1 = 38.

Now we can substitute a_1 = 38 back into the formula for the sum:

S_8 = 4(2(38) - 35)
= 4(76 - 35)
= 4(41)
= 164.

Therefore, the sum of the first 8 terms in the arithmetic sequence is 164.