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Difference between revisions of "1996 AHSME Problems/Problem 24"
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==Problem 24==
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==Problem==
  
 
The sequence <math> 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots </math> consists of <math>1</math>’s separated by blocks of <math>2</math>’s with <math>n</math> <math>2</math>’s in the <math>n^{th}</math> block. The sum of the first <math>1234</math> terms of this sequence is
 
The sequence <math> 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots </math> consists of <math>1</math>’s separated by blocks of <math>2</math>’s with <math>n</math> <math>2</math>’s in the <math>n^{th}</math> block. The sum of the first <math>1234</math> terms of this sequence is
  
 
<math> \text{(A)}\ 1996\qquad\text{(B)}\ 2419\qquad\text{(C)}\ 2429\qquad\text{(D)}\ 2439\qquad\text{(E)}\ 2449 </math>
 
<math> \text{(A)}\ 1996\qquad\text{(B)}\ 2419\qquad\text{(C)}\ 2429\qquad\text{(D)}\ 2439\qquad\text{(E)}\ 2449 </math>
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==Solution==
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The sum of the first <math>1</math> numbers is <math>1</math>
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The sum of the next <math>2</math> numbers is <math>2 + 1</math>
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The sum of the next <math>3</math> numbers is <math>2 + 2 + 1</math>
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In general, we can write "the sum of the next <math>n</math> numbers is <math>1 + 2(n-1)</math>", where the word "next" follows the pattern established above.
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Thus, we first want to find what triangular numbers <math>1234</math> is between.  By plugging in various values of <math>n</math> into <math>f(n) = \frac{n(n+1)}{2}</math>, we find:
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<math>f(50) = 1275</math>
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<math>f(49) = 1225</math>
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Thus, we want to add up all those sums from "next <math>1</math> number" to the "next <math>49</math> numbers", which will give us all the numbers up to and including the <math>1225^{th}</math> number.  Then, we can manually tack on the remaining <math>2</math>s to hit <math>1234</math>.
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We want to find:
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<math>\sum_{n=1}^{49} 1 + 2(n-1)</math>
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<math>\sum_{n=1}^{49} 2n - 1</math>
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<math>\sum_{n=1}^{49} 2n - \sum_{n=1}^{49} 1</math>
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<math>2 \sum_{n=1}^{49} n - 49</math>
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<math>2\cdot \frac{49\cdot 50}{2} - 49</math>
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<math>49^2</math>
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<math>2401</math>
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Thus, the sum of the first <math>1225</math> terms is <math>2401</math>.  We have to add <math>9</math> more <math>2</math>s to get to the <math>1234^{th}</math> term, which gives us <math>2419</math>, or option <math>\boxed{B}</math>.
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Note:  If you notice that the above sums form <math>1 + 3 + 5 + 7... + (2n-1) = n^2</math>, the fact that <math>49^2</math> appears at the end should come as no surprise.
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==Solution 2 (Alcumus) ==
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The <math>k</math>th appearance of 1 is at position <math>1 + 2 + \dots + k = \frac{k(k + 1)}{2}</math>. Then there are <math>k</math> 1's and <math>\frac{k(k + 1)}{2} - k = \frac{k(k - 1)}{2}</math> 2's among the first <math>\frac{k(k + 1)}{2}</math> numbers, so the sum of these <math>\frac{k(k + 1)}{2}</math> terms is <math>k + k(k - 1) = k^2</math>.
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When <math>k = 49</math>, <math>\frac{k(k + 1)}{2} = 1225</math>, and when <math>k = 50</math>, <math>\frac{k(k + 1)}{2} = 1275</math>.
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The sum of the first 1225 terms is <math>49^2 = 2401</math>. The numbers in positions 1226 through 1234 are all 2's, so their sum is <math>(1234 - 1226 + 1) \cdot 2 = 18</math>. Therefore, the sum of the first 1234 terms is <math>2401 + 18 = \boxed{2419}</math>.
  
 
==See also==
 
==See also==
 
{{AHSME box|year=1996|num-b=23|num-a=25}}
 
{{AHSME box|year=1996|num-b=23|num-a=25}}
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{{MAA Notice}}

Latest revision as of 19:39, 2 November 2023

Problem

The sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2,\ldots$ consists of $1$’s separated by blocks of $2$’s with $n$ $2$’s in the $n^{th}$ block. The sum of the first $1234$ terms of this sequence is

$\text{(A)}\ 1996\qquad\text{(B)}\ 2419\qquad\text{(C)}\ 2429\qquad\text{(D)}\ 2439\qquad\text{(E)}\ 2449$

Solution

The sum of the first $1$ numbers is $1$

The sum of the next $2$ numbers is $2 + 1$

The sum of the next $3$ numbers is $2 + 2 + 1$

In general, we can write "the sum of the next $n$ numbers is $1 + 2(n-1)$", where the word "next" follows the pattern established above.

Thus, we first want to find what triangular numbers $1234$ is between. By plugging in various values of $n$ into $f(n) = \frac{n(n+1)}{2}$, we find:

$f(50) = 1275$

$f(49) = 1225$

Thus, we want to add up all those sums from "next $1$ number" to the "next $49$ numbers", which will give us all the numbers up to and including the $1225^{th}$ number. Then, we can manually tack on the remaining $2$s to hit $1234$.

We want to find:

$\sum_{n=1}^{49} 1 + 2(n-1)$

$\sum_{n=1}^{49} 2n - 1$

$\sum_{n=1}^{49} 2n - \sum_{n=1}^{49} 1$

$2 \sum_{n=1}^{49} n - 49$

$2\cdot \frac{49\cdot 50}{2} - 49$

$49^2$

$2401$

Thus, the sum of the first $1225$ terms is $2401$. We have to add $9$ more $2$s to get to the $1234^{th}$ term, which gives us $2419$, or option $\boxed{B}$.

Note: If you notice that the above sums form $1 + 3 + 5 + 7... + (2n-1) = n^2$, the fact that $49^2$ appears at the end should come as no surprise.

Solution 2 (Alcumus)

The $k$th appearance of 1 is at position $1 + 2 + \dots + k = \frac{k(k + 1)}{2}$. Then there are $k$ 1's and $\frac{k(k + 1)}{2} - k = \frac{k(k - 1)}{2}$ 2's among the first $\frac{k(k + 1)}{2}$ numbers, so the sum of these $\frac{k(k + 1)}{2}$ terms is $k + k(k - 1) = k^2$.

When $k = 49$, $\frac{k(k + 1)}{2} = 1225$, and when $k = 50$, $\frac{k(k + 1)}{2} = 1275$.

The sum of the first 1225 terms is $49^2 = 2401$. The numbers in positions 1226 through 1234 are all 2's, so their sum is $(1234 - 1226 + 1) \cdot 2 = 18$. Therefore, the sum of the first 1234 terms is $2401 + 18 = \boxed{2419}$.

See also

1996 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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