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

Step-by-step explanation:

Given;

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

30

∑  (−3i+5)                    -------------------(i)

i=1

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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;

$s_n = \frac{n}{2}[a_1 + a_n]$   -----------------(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;

$s_n = \frac{30}{2}[2 + (-85)]$

$s_n$ = 15[-83]

$s_n$ = -1245