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Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

Adding an Arithmetic Series by Pairing Terms

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 2 min 18 s
  • British Columbia
  • BC10-MA-23

Pairing the first and last terms of 1 + 2 + ... + 100 gives 50 pairs that each make 101, so the sum is 5050, which leads to the general formula Sₙ = n(t₁ + tₙ) ÷ 2. The formula is then used on 7 + 11 + ... + 83, where the video shows that 19 steps of 4 join 20 terms (not 19) to get 20 × 90 ÷ 2 = 900, and checks it with 10 pairs of 90.

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In one sentence

The question this lesson answers

What is the sum of every whole number from 1 to 100?

Adding up the terms of a sequence gives a series.

Transcript

What the lesson says

What is the sum of every whole number from 1 to 100?

Adding up the terms of a sequence gives a series.

The terms go up by 1 each time, a common difference of 1, so this is an arithmetic series.

Pair the first term with the last: 1 plus 100 is 101.

Then the second with the second-last: 2 plus 99 is also 101.

3 plus 98 is 101 too. Every pair makes 101, down to the middle pair, 50 plus 51.

100 terms make 100 divided by 2, or 50 pairs.

50 pairs of 101 is 50 times 101, which is 5050.

In general, n terms make n divided by 2 pairs, each worth first plus last. So S sub n is n times t sub 1 plus t sub n, divided by 2.

Here n is the number of terms, t sub 1 is the first term, and t sub n is the last term.

For 1 to 100, S sub 100 is 100 times 101, divided by 2, which is 5050 again.

Now add 7 plus 11 plus 15, and so on, up to 83. The common difference is 4.

Count the terms with care. 83 minus 7 is 76, and 76 divided by 4 is 19 steps.

But 19 steps join 20 terms, just as 2 steps join 7, 11 and 15. So n is 20, not 19.

So S sub 20 is 20 times, 7 plus 83, divided by 2. That is 20 times 90 divided by 2, or 900.

Check with pairs: 20 terms make 10 pairs of 90, which is 900.

So, the sum is the number of terms times first plus last, divided by 2.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Linear relations: arithmetic sequencesBC10-MA-23

Previous: Arithmetic Sequences Are Linear: The Common Difference Is the SlopeNext: Solving a System by Graphing: The Point Both Lines Share

The outcomes and strands follow the British Columbia curriculum.

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