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03 Apr 2020, 02:52
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
49% (02:23) correct
51%
(02:26)
wrong
based on 115
sessions
History
Date
Time
Result
Not Attempted Yet
[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 : \(1 + 2 + … + n = \frac{n(n+1)}{2}, 1^2 + 2^2 + … + n^2 = \frac{n(n+1)(2n+1)}{6}\))
A. \(25\)
B. \(125\)
C. \(225\)
D. \(325\)
E. \(425\)
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 : \(1 + 2 + … + n = \frac{n(n+1)}{2}, 1^2 + 2^2 + … + n^2 = \frac{n(n+1)(2n+1)}{6}\))
A. \(25\)
B. \(125\)
C. \(225\)
D. \(325\)
E. \(425\)
03 Apr 2020, 06:15
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MathRevolution
Formula
Number of cards = 1 of 1 + 2 of 2 + 3 of 3+......+ n of n = \(1+2+3+...+n=\frac{n(n+1)}{2}\)
Sum of all cards = \(1*1+2*2+3*3+...n*n=1^2+2^2+3^2+...+n^2=\frac{n(n+1)(2n+1)}{6}\)
Now \(Average*Number_ of_ cards=Sum_ of_ all_ cards..........17*\frac{n(n+1)}{2}=\frac{n(n+1)(2n+1)}{6}=\frac{n(n+1)(2n+1)}{2*3}........17=\frac{2n+1}{3}......51=2n+1.......n=25\)
Number of cards = \(\frac{n(n+1)}{2}=\frac{25*26}{2}=325\)
Use choices
If you know that the Number of cards = \(1+2+3+...+n=\frac{n(n+1)}{2}\), then our answer*2 should be equal to n(n+1), that is sum of consecutive integers.
Let us check choices
A. \(25\), so 25*2=5*5*2..We cannot form this as a product of consecutive integers.
B. \(125\), so 125*2=5*5*5*2..We cannot form this as a product of consecutive integers.
C. \(225\), so 225*2=5*5*3*3=15*15....We cannot form this as a product of consecutive integers.
D. \(325\), so 325*2=5*5*13*2=25*26..YES
E. \(425\), so 425*2=5*5*17*2..We cannot form this as a product of consecutive integers.
D
06 Apr 2020, 03:55
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=>
\(\frac{( 1*1 + 2*2 + 3*3 + … +n*n ) }{ ( 1 + 2 + 3 + … + n )} = 17\)
\([(\frac{1}{6})n(n+1)(2n+1)] / [(\frac{1}{2})n(n+1) ] = \frac{(2n+1)}{3} = 17.\)
Thus, we have \(2n+1 = 51\) or \(n = 25.\)
Then the number of cards is
\(1 + 2 + 3 + … + 25 = (\frac{1}{2})25*26 = 325\)
Therefore, D is the answer.
Answer: D
\(\frac{( 1*1 + 2*2 + 3*3 + … +n*n ) }{ ( 1 + 2 + 3 + … + n )} = 17\)
\([(\frac{1}{6})n(n+1)(2n+1)] / [(\frac{1}{2})n(n+1) ] = \frac{(2n+1)}{3} = 17.\)
Thus, we have \(2n+1 = 51\) or \(n = 25.\)
Then the number of cards is
\(1 + 2 + 3 + … + 25 = (\frac{1}{2})25*26 = 325\)
Therefore, D is the answer.
Answer: D
