Question

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

0
∑
i=1 (−3i+5)

Answer 1

Question:

Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

30

∑ (−3i+5)

i=1

Answer:

The first three terms are : 2, -1 and -4

The last term is: -85

The sum of the sequence is: -1245

Explanation:

Given;

==================================

30

∑ (−3i+5) -------------------(i)

i=1

==================================

Where the ith term aₙ is given by;

`a_(i)` =
`-3i + 5` -------------------(ii)

(a) Therefore, to get the first three terms (
`a_1, a_2, a_3`), we substitute i=1,2 and 3 into equation (ii) as follows;

`a_(1)` =
`-3(1) + 5` = 2

`a_2 = -3(2) + 5 = -1`

`a_3 = -3(3) + 5`
`= -4`

Since the sum expression in equation (i) goes from i=1 to 30, then the last term of the sequence is when i = 30. This is given by;

`a_(30) = -3(30) + 5 = -85`

(b) The sum
`s_n` of an arithmetic sequence is given by;

`s_n = (n)/(2)[a_1 + a_n]` -----------------(iii)

Where;

n = number of terms in the sequence = 30

`a_1` = first term = 2

`a_n` = last term = -85

Substitute the corresponding values of n,
`a_1` and
`a_n` into equation (iii) as follows;

`s_n = (30)/(2)[2 + (-85)]`

`s_n` = 15[-83]

`s_n` = -1245