The Student Room Group

how do i answer this sequence question?

Find the sum of the multiples of 3 less than 100. Hence or otherwise find the sum of the numbers less than 100 which are not multiples of 3.
Reply 1
Original post by Harry Long
Find the sum of the multiples of 3 less than 100. Hence or otherwise find the sum of the numbers less than 100 which are not multiples of 3.

Firstly I recommend writing out the first few terms of the sequence so you have a picture of what's going on.

First question : what's the last term of the sequence? Then how can you use that to find nn i.e. the number of terms in the sequence?
Reply 2
so first few terms are:
3,6,9,12,15,18, ..., 99

number of terms = (99 - 3 / 3) + 1 = 33 terms

first is 3 and last is 99
so a = 3 and l is 99

Sn = n/2(a + l)
Sn = 33/2(3 + 99)
Sn = 1683
im guessing this is wrong?
Reply 3
i feel like ive done it a really long way and theres acutally a shorter way e.e
Reply 4
For the second question, would I have to find the sum of the first 100 numbers and then subtract 1683 (sum of multiples of 3 under 100) from that?
Reply 5
So

a = 1, l = 100
n = 100 (because 100 terms)

Sn = n/2(a + l)
Sn = 50(1 + 100)
Sn = 50 times 101
Sn = 5050

So sum of terms under 100 that are nota m ultiple of 3 = 5050 - 1683 = 3367?
Reply 6
Oops, the last question i did wrong. The actual answer is 3467 and i got 3367 :frown: Does anyone know why
Reply 7
Original post by Harry Long
Find the sum of the multiples of 3 less than 100. Hence or otherwise find the sum of the numbers less than 100 which are not multiples of 3.


You use the concept of Mathematical Induction. Essentially, it is an effective way of summarising numbers.

You can read about it here:
https://www.mathsisfun.com/algebra/mathematical-induction.html

There are other resources, so the one suggested is just an example.
Reply 8
Original post by Harry Long
Oops, the last question i did wrong. The actual answer is 3467 and i got 3367 :frown: Does anyone know why


The correct answer is 3267.

You have 3367 because you included the 100 in your answer.

The question asks: the sum of the numbers less than 100. You have included the value 100 in your answer, which is incorrect.

In short:

The sum of numbers 1 - 99, is:

99 * (99 + 1)
----------------- = 4950.
2

Then the sum of the numbers from 1 - 99 that are multiples of 3 is 1683.

4950 - 1683 = 3267
Reply 9
thanks, now i understand i just made a slight slip because i thought it included 100
Reply 10
Original post by Baleroc
You use the concept of Mathematical Induction. Essentially, it is an effective way of summarising numbers.

You can read about it here:
https://www.mathsisfun.com/algebra/mathematical-induction.html

There are other resources, so the one suggested is just an example.

Definitely no need for induction in this question.

Quick Reply