(1^2+2^2+3^3+…+20^2) / (1+2+3+ …+20)=?

OA: A

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

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

Dividing \((1)\) by \((2)\), we get

\(\frac{Sum \quad of \quad the \quad squares \quad of \quad first \quad n \quad natural \quad numbers}{Sum \quad \quad of \quad first \quad n \quad natural \quad numbers}=\frac{\frac{n(n+1)(2n+1)}{6}}{\frac{n(n+1)}{2}}=\frac{2n+1}{3}\)

Putting \(n = 20\) , we get

\(\frac{Sum \quad of \quad the \quad squares \quad of \quad first \quad 20 \quad natural \quad numbers}{Sum \quad \quad of \quad first \quad 20 \quad natural \quad numbers}=\frac{2*20+1}{3}=\frac{41}{3}\)

The sum 1+2+3+...+20 is equal to 20*(1+20)/2 = 21*10 and it is certainly in-the-GMAT-scope. (We have used a"finite arithmetic progression" important result.)

The sum of the squares of the first "n" (n=20) positive integers is out-of-GMAT´s scope, although a beautiful (and elementary) deduction for the formula:

\({1^2} + {2^2} + \ldots + {n^2} = \frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}\)

you may find here: https://mathschallenge.net/library/numb ... of_squares

From the identity above, we have:

\({1^2} + {2^2} + \ldots + {20^2} = \frac{{20 \cdot 21 \cdot 41}}{6} = 10 \cdot 7 \cdot 41\)


Finally,

\(? = \frac{{10 \cdot 7 \cdot 41}}{{21 \cdot 10}} = \frac{{41}}{3}\)


The right answer is therefore (A) .


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

P.S.: perhaps there is a smart way of avoiding the "out-of-scope formula"... some "hidden pattern", for instance...
Anyway, I could not find an alternate approach in approx. 5min (a generous "time-limit" in the GMAT).

Hi, Mitch! You had a creative idea, no doubt, but...

01. You have made CONSIDERABLE modifications in the original parcels... the alternative choices are not that apart for these modifications... far from that!
02. We are dealing with sum of SQUARES, arithmetic means (averages) are dangerous (to say the least) in nonlinear situations.

There was a "happy ending", I saw... but quoting the great Niels Bohr...

"Prediction is very difficult, especially about the future."

Regards,
Fabio.

I guess I found a nice approximation solution inspired by Mitch´s clever idea.
(If he accepts the reasoning, we share the merits!)


\({1^2} + {2^2} + \ldots + {20^2} < 10 \cdot {10^2} + 10 \cdot {20^2}\)

\(? = \frac{{{1^2} + {2^2} + \ldots + {{20}^2}}}{{10 \cdot 21}} < \frac{{{{10}^2} + {{20}^2}}}{{21}} = \frac{{500}}{{21}}\,\,\mathop \cong \limits^{\left( * \right)} \,\,24\)

\(\left( * \right)\,\,\,\frac{{500}}{{21}} = \frac{{420 + 84 - 4}}{{21}} = 23\frac{{17}}{{21}}\)


\(\left( A \right)\,\,\frac{{41}}{3} = 13\frac{2}{3}\)


Note that (C), (D) and (E) are out, while (B) is half the value of (A)... "much much less" (?) than 24...


Obs.: we may divide the sum of squares into FOUR parcels (not TWO, as before) to MAJORATE (first edited) and also MINORATE (second edited) our numerator-focus with better precision:

\(\frac{{5\left( {{3^2} + {8^2} + {{13}^2} + {{18}^2}} \right)}}{{10 \cdot 21}}\,\,\, < \,\,\,?\,\,\, < \,\,\,\frac{{5\left( {{5^2} + {{10}^2} + {{15}^2} + {{20}^2}} \right)}}{{10 \cdot 21}}\)

\(13\frac{{10}}{{21}}\,\, = \,\,\frac{{283}}{{21}}\,\,\, < \,\,\,?\,\,\, < \,\,\,\frac{{125}}{7}\,\, = \,\,17\frac{6}{7}\)


Now we are sure (A) is the right answer... and this COULD be done in less than 5min!

Regards,
Fabio.

The answer choices represent the ratio between the numerator and the denominator of the given expression.
Option A implies that the numerator is about 14 times the denominator
Option B implies that the numerator is less than 7 times the denominator.
Option C implies that the numerator is 41 times the denominator.
Option D implies that the numerator is 210 times the denominator.
Option E implies that the numerator is 420 times the denominator.
The ratios implied by the answers choices are quite different.
The estimations in my earlier post are sufficient to determine whether the numerator is less than 7 times the denominator, about 14 times the denominator, 41 times the denominator, 210 times the denominator, or 420 times the denominator.

Consider the extremes:
If the 11 greatest terms in the numerator were all 10²=100, the numerator = 11(100) + (sum of the 9 smallest terms) ≈ 1100 + 200 = 1300.
In this case, numerator/denominator ≈ 1300/200 = 6.5.
If the 11 greatest terms in the numerator were all 20²=400, the numerator = 11(400) + (sum of the 9 smallest terms) ≈ 4400 + 200 = 4600.
In this case, numerator/denominator ≈ 4600/200 = 23.
Thus, the correct ratio must be between 6.5 and 23 -- more specifically, somewhere in the MIDDLE of this range.
Only option A is viable.

=>

Now, \(1^2 + 2^2 + 3^2 + … + n^2 = \frac{n(n+1)(2n+1)}{6}\) and \(1 + 2 + 3 + … + n = \frac{n(n+1)}{2}\). So,
\(\frac{(1^2+2^2+3^2+…+20^2)}{(1+2+3+ …+20)} = ( 20*21*\frac{41}{6} ) / ( 20*\frac{21}{2} )\\
= (\frac{41}{6}) / (\frac{1}{2}) = \frac{41}{3}\)

Therefore, the answer is A.

Answer: A

A different approach would be to find a certain pattern. This one definitly takes more time, but it worked for me.

\(\frac{1^2+2^2}{1+2}\) = \(\frac{5}{3}\)

\(\frac{1^2+2^2+3^2}{1+2+3}\) = \(\frac{14}{6}\) = \(\frac{7}{3}\)

\(\frac{1^2+2^2+3^2+4^2}{1+2+3+4}\) = \(\frac{30}{10}\) = \(3\)

\(\frac{1^2+2^2+3^2+5^2}{1+2+3+4+5}\) = \(\frac{55}{15}\) = \(\frac{11}{3}\)

One can see that the denominator of the simplified fractions follows a certain pattern --> Number of terms in the denominator multiplied with 2 plus 1.
One can see that the numerator of the simplified fractions follows a certain pattern --> Always 3

As a result \(\frac{(1^2+2^2+3^2+…+20^2)}{(1+2+3+ …+20)}\) can be written as \(\frac{20*2+1}{3}\) = \(\frac{41}{3}\)

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