Chapter 2 · Sequences · Concept 1
2-4 Sum of Arithmetic Sequence (1)
Adding a hundred numbers by hand takes an afternoon and one mistake ruins it. There is a way to do it in a single line — and the trick behind it is one that a schoolboy is said to have found while being punished with exactly that homework.
Guiding question
Is there a way to add a long list without adding it?
Learning outcome. Find the sum of an arithmetic sequence, from the first term or from any term to any other.
Two names, then two formulas
In a sequence with n terms, n is the number of terms and the final one is the last term, written l. We write Sₙ for the sum of the first n terms.
when you know the last term
Sₙ = ½n(a + l)
when you know the difference
Sₙ = ½n{2a + (n − 1)d}
They are the same formula. Put l = a + (n − 1)d into the first one and you get the second. Use whichever matches what the question handed you.
If you do not know how many terms there are:
l = a + (n − 1)d
Find n from that first, then do the sum. Never guess the count.
Write the sum out. Then write it out again backwards, underneath. Now look down the columns — every single one adds up to the same thing.
| S = | 3 | 7 | 11 | 15 | 19 | 23 |
Why every column is equal. Step one place right along the top row and you gain d. Step one place right along the bottom row and you lose d, because it is running the other way. The two cancel, so the column total never moves.
There are n columns, each worth a + l. But you wrote the sum twice, so that total is 2S. Halve it, and you have the formula.
1(b) First term 2, common difference 3, last term 47. Find the sum.
- 1
Notice what is missing.
you know a, d and l — but not nBoth sum formulas need n. So n is the first job, not the sum.
“Find the sum from the 10th term to the 20th term.” The idea is right: take the whole thing and subtract the part you do not want. The question is where to cut.
sum from term p to term q = Sq − Sp−1
For 90, 84, 78, … find the sum from the 10th term to the 20th term.
S₂₀ − S₉ = 660 − 594 = 66
Read it as: throw away everything BEFORE where you want to start. To start at term 10, throw away terms 1 to 9. Subtracting S₁₀ throws away term 10 as well — the very term the question asked you to include.
Work with a partner. Before you use a sum formula, both say out loud what n is. Most wrong answers on this page are a wrong n, not wrong arithmetic.
1. Find the sum.
For the arithmetic sequence 94, 89, 84, …, find the sum of the odd terms from the first term to the 11th term.
“Odd terms” means the ones in odd positions — the 1st, 3rd, 5th and so on. Not the terms whose value happens to be an odd number.
In Try 2 the sum from the 20th to the 30th term came out as exactly 0, even though eleven real numbers were added. Could that ever happen with a sequence whose common difference is positive and whose first term is also positive?
Think about what has to be true for a total of positive numbers to be zero.
Subtracting the wrong S in a part sum.
From term 10 to term 20 is S₂₀ − S₉. Subtracting S₁₀ deletes the 10th term, which was asked for.
Guessing n instead of finding it.
If the question gives the last term but not the count, use l = a + (n − 1)d first. Every later number depends on that n.
Using ½n(a + l) when l is not the n-th term.
The l in that formula must be the LAST term of the stretch you are adding — not any term that happens to be lying around.
Sign slips with a negative d.
2a + (n − 1)d with d = −5 means subtracting. Put the brackets in: {2(20) + 12(−5)}. Rushing this line is the commonest arithmetic error in the chapter.
Leaving Sₙ untidied.
½n(4n − 2) is correct but not finished. The book wants 2n² − n.
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