Find the sum: 12 + 5 + (-2) + (-9) + ... + (-58).

Question

Find the sum: 12 + 5 + (-2) + (-9) + ... + (-58). Make a rule for the sequence and use it to figure out how many terms there are. Then use the formula for the sum.

Answer 1

The common difference is -7

Let n = number of terms

12 - 7(n-1) = -58

-7(n-1) = -70

-7n = -77

n = 11 (11 terms)

Formula for k^th term:

a_k = 12 - 7(k-1), k= 1,2,..., 11

Sum = (n/2)(2a₁ + (n-1)d) where a₁ = first term, d = common difference, n=number of terms

= (11/2)(2(12) + (11-1)(-7))

= (11/2)(24 + (-70))

= (11/2)(-46)

= -253



Answer 2

Each term is 7 less than the term before it, so this is an arithmetic sequence. The rule (formula) is:

a_n = 12 - 7(n-1) for n = 1, 2, 3, ...

where a_n is the nth term in the sequence. The sum of the first n terms of an arithmetic sequence is given by the formula:

sum = n(a₁+a_n)/2

So we need to find n when a_n = -58:

-58 = 12 - 7(n-1)

-70 = -7(n-1)

10 = n - 1

11 = n

sum = 11(12-58)/2 = ____?