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How to find the sum of a series between the 10th and 20th term inclusive?

Hi,
How do you find the sum of an arithmetic series when it is between the 10th and 20th terms? The series I have in mind has first term =2 and common difference =2. The 10th term is 20 and the 20th term is 40. Given this, I worked out that n=20, because 40 = 2+2(n-1), which is the formula for the last term.

With this in mind, using the standard sum formula, S=0.5n(2a+(n-1)d, I get the result that S=420. This is wrong, however, and the actual answer is 330. Why is this? I think it must be to do with n, given the fact that it is the 10th and 20th term inclusive.

Reply 1

niamhdoesmaths

You want the sum of the 10th - 20th terms, what you’ve calculated is the first 20 terms. You therefore don’t want the first 9 terms, so you subtract the sum of the first 9 terms from the sum of the first 20, to get the sum of the 10th - 20th :smile:

Reply 2

the bear

you can set up a new sequence where the starting term ( = a ) is the old 10th term. then the final term will be the 11th term. the value of d will be the same ?

Reply 3

Aayush :)

Person above is correct. A tip i'd suggest to be a bit faster is to set the 10th term as your a term and use n = 11. This is effectively shifting the sequence.

(edited 4 years ago)

Reply 4

Aayush :)

Original post by the bear

you can set up a new sequence where the starting term ( = a ) is the old 10th term. then the final term will be the 11th term. the value of d will be the same ?

Just beat me to that XD

Reply 5

the bear

Original post by Aayush :)

Just beat me to that XD

Reply 6

peterdxherty

Y'all, I found an easier way that I think works generally. That is, seeing as I need to find a value of n, and I already know that it's between (and inclusive of) the 10th and 20th terms, 20-10+1 = 11, and this gives the correct answer 330.

Reply 7

niamhdoesmaths

Original post by Aayush :)

Person above is correct. A tip i'd suggest to be a bit faster is to set the 10th term as your a term and use n = 11. This is effectively shifting the sequence.