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the value of $51 + 52 + 53 + \cdots + 80$

I know how to figure out questions using the formula $S= \frac{n(n + 1)}{2}$. The sum being $S$ and first positive integers being $n$.

But the numbers in the equation aren't the first. How do i figure this question out? Do i have to change the equation?

2 Answers 2

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You don't have to change the equation, but you do have to use it in a creative way. You have two options:

One, you can rewrite this as

$$(50+1) + (50+2) + \cdots + (50+30)$$

which is the same as

$$50+50+50+\cdots + 50 + 1 + 2 + \cdots + 30$$

The other, you can say that $$51+52+\cdots + 80 = (1+2+\cdots + 80) - (1+2+\cdots + 50)$$

Both options should give you the same result.

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I have figured it out thanks to the first answer. Whoever is looking at this needing help you just have to figure out the sum of the first 80 positive integers and then minus that by the sum of the first 50 integers. so your working out should look something like this. 80/2(80+1) - 50/2(50+1) which then gives you 1965.

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    You know, if my answer helped you solve the problem, then the typical thing to do is to accept the answer... – 2017-02-20
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    It wasnt your answer, it was someone else's but they seem to have deleted it. – 2017-02-26