Calculate 2, 5, 8, 11, ..., sum of the first 10 terms Step by Step

Step by step

The question

Question: 2, 5, 8, 11, …: find the sum of the first 10 terms.

A sequence is a list of numbers in order.

2, 5, 8, 11, ...

1 Find the difference

We add 3 every time.

A list like this is an arithmetic sequence.

The number we add is called the difference: d = 3.

d = 3

2 Find the last term

The last term is term 10.

To get there, we add 3, 9 times.

a₁₀ = 2 + (10 − 1) × 3

Brackets come first.

Order: ( ) → x² √ → × ÷ → + −

10 minus 1 is 9.

= 2 + 9 × 3

Times comes before plus.

9 times 3 is 27.

= 2 + 27

2 plus 27 is 29.

a₁₀ = 29

a₁ is term 1, a₂ is term 2.

So aₙ is term number n.

a_n = term number n

3 Add in pairs

S is the sum: all 10 terms added.

Now the same sum backwards: the last term first.

Add each column.

2 + 29 is 31.

5 + 26 is 31.

Every column makes 31!

There are 10 columns.

So the two sums make 10 times 31.

That is 310.

2S = 10 × 31 = 310

One sum is half of that.

We divide by 2.

S = 3102

So the sum of the first 10 terms is 155.

S₁₀ = 155

This works for every arithmetic sequence: the number of terms, times first plus last, over 2.

S_n = n(a₁ + a_n)2

4 Check

Let's check with the rule.

S₁₀ = 10(2 + 29)2 = 10 × 312 = 155

We get 155 again.

So the sum is right!

Let's look back.

First: find the difference.

Then: find the last term.

Then: add in pairs.

Last: check.

Answer