# Find the sum, the sum of the squares, and the sum of the cubes for the first $250$ natural numbers.

1. $784,375$
2. $\frac{250(251)(501)}{6} = 5,239,625$
3. $\frac{250^2(251)^2}{4} = 984,390,625$

How is the first answer $784,375?$ The answer I get is $31,375$.

• More pertinently, why do you say that the answer to the sum of the numbers from $1$ to $250$ is $784,375$ when that is manifestly far too large? – Angina Seng Feb 24 '18 at 17:40
• He/she did not obtain that answer, the answer provided, I believe is incorrect. – user532449 Feb 24 '18 at 17:41
• 784,375 is the given answer. Any chance we are misinterpreting the question? – user534801 Feb 24 '18 at 17:47
• What is the exact question? – user532449 Feb 24 '18 at 18:18
• "Find the sum, the sum of the squares, and the sum of the cubes for the first 250 natural numbers." – user534801 Feb 24 '18 at 18:29

The sum of an AP consisting of $n$ terms is given by

$\dfrac{n(a_1+a_n)}{2}$

Here, the sum of natural numbers till 250 is $\dfrac{250(251)}{2}= 31375$

The first $250$ natural numbers are $1-250$.

Finding the average of $1$ and $250$, you get $125.5$

Since the average is $125.5$, you multiply $125.5$ by $250$, since you are accounting for the first $250$ natural numbers.

$125.5*250=31,375$

The answer provided ($784,375$) is incorrect.

• Note the average of the largest and smallest numbers in a set if numbers is not necessarily the average of all of the numbers. If the numbers are an arithmetic sequence, as is the case here, then that is correct. Yolo Inver gives a very quick and nice answer. – user247327 Feb 24 '18 at 17:48
• @user247327 Thank you for pointing that out! – user532449 Feb 24 '18 at 17:54
• I guess this was downvoted, since it was not a useful answer. – user532449 Feb 24 '18 at 19:02

Write the first 250 natural numbers as

1 2 3 4 5 ... 248 249 250 and below then write

250 249 248 247 246 ... 3 2 1

Adding each pair vertically gives 251 for each pair. There are 250 pairs so the total is (250)(251)= 62750. But each number from 1 to 250 has been added twice. The sum of the first 250 natural numbers is half of that: 62750/2= 31375. Where did you get "748375"?