Sum from i = 3 to 7 of (2i - 5) Step by Step

Step by step

The question

Question: work out the sum for i = 3 to 7 of (2i − 5).

Σ_{i = 3}⁷(2i − 5)

1 Read the sigma

Σ is the Greek letter sigma.

It means: add up.

Start with i = 3, and stop at i = 7.

That is 7 − 3 + 1: 5 terms.

For each i we work out 2i − 5.

Σ = add up

start: i = 3, stop: i = 7

7 − 3 + 1 = 5 terms

Careful!

From 3 to 7 there are 5 terms, not 4: we count both ends.

So 7 − 3 + 1.

Careful: 7 − 3 = 4 is not the number of terms

2 Write the terms

i = 3: we put 3 in place of i.

We get 1.

i = 3: 2(3) − 5 = 6 − 5 = 1

i = 4 gives 3.

i = 4: 2(4) − 5 = 8 − 5 = 3

i = 5: 2(5) − 5 = 10 − 5 = 5

i = 6: 2(6) − 5 = 12 − 5 = 7

And i = 7 gives 9.

That is the last one.

i = 7: 2(7) − 5 = 14 − 5 = 9

3 Add them

Now we add the 5 terms.

Σ_{i = 3}⁷(2i − 5) = 1 + 3 + 5 + 7 + 9

So the sum is 25.

= 25

4 Check

Let's check with the pairs from the sequences lesson.

The terms go up by the same amount, so first plus last, 1 + 9, times 5 terms, over 2.

Pairs: 5 × (1 + 9)2 = 5 × 102 = 25

We get 25 again.

It works!

Let's look back.

First: read the sigma.

Then: write the terms.

Then: add them.

Last: check.

Answer