Problem Solving

[GMAT math practice question]

There are 1 card written ‘1’, 2 cards written ‘2’, …, and n cards written ‘n’. The average of all the numbers written on the cards is 17. How many cards are there? (Use the fact : 1 + 2 + … + n = n(n+1)/2, 1^2 + 2^2 + … + n^2 = n(n+1)(2n+1)/6)

A. 25
B. 125
C. 225
D. 325
E. 425

we have a sequence of card in which
1 card has '1' written on it
2 cards have '2' written on it
3 cards will have '3' written on it
n cards will have 'n' written on it
Total no of cards = $$1+2+3+n=\frac{n\left(n+1\right)}{2}$$
And sum of all numbers on the card
$$1^2+2^2+3^2+n^2=$$
$$\frac{n\left(n+1\right)\left(2n+1\right)}{6}$$
Given that average of all numbers written on the card = 17 and the total number of cards= ?
and average = $$\frac{Sum\ of\ numbers\ on\ card}{Total\ number\ of\ cards\ }$$
$$17=\frac{n\left(n+1\right)\left(n+2\right)}{6}\cdot\frac{2}{n\left(n+1\right)}$$ $$17=\frac{n\left(n+1\right)\left(2n+1\right)}{3n\left(n+1\right)}$$ $$17\cdot3=\left(2n+1\right)$$ $$51=\left(2n+1\right)$$ $$\frac{50}{2}=\frac{\left(2n\right)}{2}$$ $$n=25$$ $$n=25$$
Total number of cards = n(n+1)/2 where n=25
$$\frac{25\left(25+1\right)}{2}=\frac{25\cdot26}{2}=\frac{650}{2}=325$$ $$Answer\ is\ Option\ D$$

=>

( 1*1 + 2*2 + 3*3 + … +n*n ) / ( 1 + 2 + 3 + … + n ) = 17
[(1/6)n(n+1)(2n+1)] / [(1/2)n(n+1) ] = (2n+1)/3 = 17.
Thus, we have 2n+1 = 51 or n = 25.

Then the number of cards is
1 + 2 + 3 + … + 25 = (1/2)25*26 = 325

Therefore, D is the answer.
Answer: D