Chapter 2 · Sequences · Concept 1
2-5 Sum of Arithmetic Sequence (2)
There is no new formula in this lesson. Not one. What is new is that nobody hands you the first term any more — the question says something like 'the multiples of 3 among the two-digit numbers' and expects you to work out what that even means.
Guiding question
How do you turn a sentence about numbers into a sequence you can add?
Learning outcome. Use arithmetic sequences to find sums of natural numbers and of multiples.
Same formulas, new job
with the last term
Sₙ = ½n(a + l)
with the difference
Sₙ = ½n{2a + (n − 1)d}
The only job in this lesson:
find the first, the last, and how many
Once you have those three numbers, the question is finished — the formula does the rest. And when the count is not obvious, get it from l = a + (n − 1)d rather than guessing.
Pick a sentence and watch it turn into a first term, a last term and a count. Notice how often the first term is not the number in the sentence.
Find the sum of the multiples of 3 among the 2-digit numbers.
- press the button to translate it
2-digit means 10 to 99. The first multiple of 3 in there is 12, not 10.
How many numbers from 101 to 200?
It is tempting to say 200 − 101 = 99. That is the gap, not the count.
200 − 101 + 1 = 100
Test it on something small: from 3 to 5 the gap is 2, but the numbers are 3, 4, 5 — three of them. The +1 is the first number, which subtracting always forgets.
How many multiples of 3 from 10 to 99?
Do not divide and hope. Find the real first and last, then use the formula.
first 12, last 99, step 3
99 = 12 + (n − 1) × 3
87 = 3(n − 1) → n − 1 = 29 → n = 30
Thirty, not twenty-nine. The formula does the +1 for you, which is exactly why it is safer than counting in your head.
3. Find the sum of the multiples of 3 among the 2-digit natural numbers.
- 1
Work out what "2-digit" means as numbers.
from 10 up to 99Say it in numbers before anything else. 10 is the smallest, 99 the biggest.
Work with a partner. Before either of you writes a formula, both say the three numbers: first, last, how many. If you disagree on the count, that is the question — sort it out first.
Find the following sums.
Find the sum of the numbers from 1 to 100 that are not divisible by 4.
Exercise 9 asks for the numbers from 10 to 100 that are not divisible by 6, and the answer is 4195. Why can that one not be done straight, and how would you know to switch methods before wasting time?
Write out the first six or seven numbers and look at the gaps.
Counting with a subtraction and no +1.
From 81 to 150 there are 150 − 81 + 1 = 70 numbers. Forgetting the +1 loses one whole term and every number after it.
Starting at the number in the sentence.
"Multiples of 3 among the 2-digit numbers" starts at 12, not 10. Find the first one that actually fits the description.
Stopping short at the end.
99 IS a multiple of 3, so it belongs. Check the top of the range as carefully as the bottom.
Trying to add a set with holes in it.
"Not divisible by 4" is not an arithmetic sequence. Add everything, then subtract the multiples of 4.
Reading "2-digit" as 1 to 99.
It means 10 to 99. Single-digit numbers are not 2-digit numbers.
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