What is the average of the following numbers?

Question

What is the average of the following numbers? 
1^2, 2^2, 3^2...10^2.

A. 5.5
B. 25
C. 38.5
D. 55
E. 385

Easy problem but may take time. The book explains it the following way which I am wondering about:
“∑_(n=1)^10▒〖n^2=〗 1/6 n(n+1)(2n+1)=385. The average is therefore 385/10 = 38.5”I do not understand the bits and pieces of the above. Moreover, I am wondering if there is an alternative shortcut to solve this type of problem? 
Or, for example, is there a way to calculate the sum of 1^2 through…20^2 using the above shortcut or another method?
What is the number property tested here?

Thanks in advance.

MacFauz · Accepted Answer

Sum of n consecutive integers from 1 to n =  \frac{n(n+1)}{2} 
Sum of squares of n consecutive integers from 1 to n =  \frac{n(n+1)(2n+1)}{6} 
Sum of cubes of n consecutive integers from 1 to n =  (\frac{n(n+1)}{2})^2

Vips0000 · Answer

It is not necessary usually on GMAT to remember formulae. 
This question could be solved logically knowing number properties by process of elimination.
It is very easy to know the unit digit for each term and sum of it (1+4+9+6+5+9+4+1+0) = sum of unit digits end with a 5. Also total number of terms are 10. So any number ending with 5 when divided by 10 can not be a whole number. Thus we narrowed down to A and C. Now A is only 5.5 which signifies that sum of all number must be 55 which is incorrect as it is clearly less than even a single term 10^2 (or 100). 
So we get remaining choice C as correct answer.

pritish2301 · Answer

Hi Vips,

The strategy applied by you will work if the answer choices are not similar for example
A. 39.5
B. 38.5

Are such questions common in Gmat?

P.S: My apporach to solution was just like yours

Thanks,
Pritish

Vips0000 · Answer

Hi Pritish,

Truely speaking, from what I've seen and what I've heard 'experts' say is that GMAT doesnt test number crunching but the application of knowledge. For the given problem we were able to solve easily using number properties. if the numbers in options are closer as u mentioned, usually calculation would still be minimal on actual gmat problem.

craky · Answer

My biggest problem is that sometime I dont see the easy solutions and end up wasting my time on prolonged calculations. Shortcuts are always useful. And I would also say that GMAT usually provides reasonably different answer choices so that shortcuts tend to work fine.

pritish2301 · Answer

Hi Craky,
I had the same problem too. Reading one of gmat blogs I understood that gmat will not test your calculation ability (how fast we can add or multiply numbers) rather it tests if we can apply any learnt math priniciple here.

In case you come across such questions, please feel free to share in the forum.

Some instances :
1. A problem where applying (a+b)(a-b) rule gave the answer quickly
2. Take out the common terms and break the numbers

Regards,
Pritish

GGMAT760 · Answer

Hi.. Can anyone tell me, what does n mean in the equations? n = total number????? or first term????

PareshGmat · Answer

n is the last term

In this case, n=10

So  \frac{10 * 11 * 21}{6} = 385

Average  = \frac{385}{10} = 38.5

KrishnakumarKA1 · Answer

sum of square of number from 1,2,3...n = {n(n+1)(2n+1)}/6
average = {n(n+1)(2n+1)}/6 * 1/n = {(n+1)(2n+1)}/6

putting n =10 in the above expression we get
{(10+1)(2*10+1)}/6
= 11*21/6
= 38.5

Option C

Abhishek009 · Answer

Sum of the squares of first n natural numbers =  \frac{n(n+1)(2n+1)}{6}

So, Sum of the squares of first 10 natural numbers =  \frac{10*11*21}{6}  =  385

Average of these 10 numbers is 38.50, Answer must be (C)

AkshdeepS · Answer

A. 5.5
B. 25
C. 38.5
D. 55
E. 385

As the answer choices are quite spreaded, we can use approximation.

(1 + 4)  +  (9 + 16)  +  (25 + 36)  +  49  +  (64 + 81)  +  100

We can add them roughly

5+25+60+50+150+100

390/10 = 38.5 approx.

(C)

KarishmaB · Answer

Yes, even if you do not know the formula, a bit of logic and number properties can help you solve the question. The numbers will be

1, 4, 9, 16, 25, 36, 49, 64, 81, 100 
Note that as you go to higher numbers, the square spreads out more. the difference between 8^2 and 9^2 will be much more than the difference between 6^2 and 7^2.

Only 3 numbers are greater than 50 and that too not all very far from it but we do have a lot of very small numbers (1, 4, 9) so the average is certainly less than 50.

In increasing order, 30 lies about mid way (between 25 and 36) but deviation of the numbers to the right of it is much more than the deviation to the left of it. Hence, the average should certainly be greater than 25.

Answer (C)

el2010 · Answer

Great alternative approach. Very insightful. Thank you.

ScottTargetTestPrep · Answer

The average is:

(1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100)/10

385/10 = 38.5

Answer: C

jcerdae · Answer

Thank you for the great solutions and formulas!

Is there a formula if the first consecutive number is not 1?
What if (for example) the question says:

What is the average of the following numbers?
11^2, 12^2, 13^2...20^2.

---------
What I would do is calculate from 1^2 to 10^2 and from 1^2 to 20^2. and then substract

Do you think the GMAT can ask this or this type of questions always start with 1 as the first digit?
Thank you!!

Kinshook · Answer

Asked: What is the average of the following numbers? 
1^2, 2^2, 3^2...10^2.
Sum of {1^2, 2^2, 3^2...10^2} = n(n+1)(2n+1)/6 = 10*11*21/6 = 5*11*7 = 385
Average of {1^2, 2^2, 3^2...10^2} = 385/10 = 38.5

IMO C

ckavishwar · Answer

We know, Average of a series = (Sum of all Numbers) / (Total Quantity of Numbers)

As sum of 1st n square numbers = {n(n+1)(2n+1)}/6

Hence average = {n(n+1)(2n+1)}/6*n ; where n = quantity of numbers

As n=10 ; therefore avg = {10*(10+1)(2*10 + 1)} / (6 * 10)

= 10*11*21 / 60

= 38.5 ; Hence C

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What is the average of the following numbers?

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