Difference between revisions of "1985 AJHSME Problems/Problem 2"

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<math>\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045</math>
 
<math>\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045</math>
  
==Solution==
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==Solution 1==
 
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One possibility is to simply add them. However, this can be time-consuming, and there are other ways to solve this problem.  
===Solution 1===
 
We could just add them all together. But what would be the point of doing that? So we find a slicker way.
 
 
 
 
We find a simpler problem in this problem, and simplify -> <math>90 + 91 + ... + 98 + 99 = 90 \times 10 + 1 + 2 + 3 + ... + 8 + 9</math>
 
We find a simpler problem in this problem, and simplify -> <math>90 + 91 + ... + 98 + 99 = 90 \times 10 + 1 + 2 + 3 + ... + 8 + 9</math>
  
We know <math>90 \times 10</math>, that's easy - <math>900</math>. So how do we find <math>1 + 2 + ... + 8 + 9</math>?
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We know <math>90 \times 10</math>, that's easy: <math>900</math>. So how do we find <math>1 + 2 + ... + 8 + 9</math>?
  
 
We rearrange the numbers to make <math>(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5</math>. You might have noticed that each of the terms we put next to each other add up to 10, which makes for easy adding. <math>4 \times 10 + 5 = 45</math>. Adding that on to 900 makes 945.
 
We rearrange the numbers to make <math>(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5</math>. You might have noticed that each of the terms we put next to each other add up to 10, which makes for easy adding. <math>4 \times 10 + 5 = 45</math>. Adding that on to 900 makes 945.
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945 is <math>\boxed{\text{B}}</math>
 
945 is <math>\boxed{\text{B}}</math>
  
===Solution 2===
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==Solution 2==
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We can express each of the terms as a difference from 100 and then add the negatives using <math>\frac{n(n+1)}{2}</math> to get the answer.
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<cmath>\begin{align*}
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(100-10)+(100-9)+\cdots + (100-1) &= 100 \times 10 -(1+2+\cdots +9+10)\\
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&= 1000 - 55\\
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&= 945 \rightarrow \boxed{\text{B}}
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\end{align*}</cmath>
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 +
==Solution 3==
 
Instead of breaking the sum and then rearranging, we can start by rearranging:
 
Instead of breaking the sum and then rearranging, we can start by rearranging:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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\end{align*}</cmath>
 
\end{align*}</cmath>
  
===Solution 3===
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==Solution 4==
  
We can use a formula.  
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We can use the formula for finite arithmetic sequences.
  
It is <math>\frac{n}{2}\times</math> (First term+Last term) where <math>n</math> is the number of terms in the sequence.  
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It is <math>\frac{n}{2}\times</math> (<math>a_1+a_n</math>) where <math>n</math> is the number of terms in the sequence, <math>a_1</math> is the first term and <math>a_n</math> is the last term.
  
 
Applying it here:
 
Applying it here:

Revision as of 02:19, 10 June 2021

Problem

$90+91+92+93+94+95+96+97+98+99=$


$\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045$

Solution 1

One possibility is to simply add them. However, this can be time-consuming, and there are other ways to solve this problem. We find a simpler problem in this problem, and simplify -> $90 + 91 + ... + 98 + 99 = 90 \times 10 + 1 + 2 + 3 + ... + 8 + 9$

We know $90 \times 10$, that's easy: $900$. So how do we find $1 + 2 + ... + 8 + 9$?

We rearrange the numbers to make $(1 + 9) + (2 + 8) + (3 + 7) + (4 + 6) + 5$. You might have noticed that each of the terms we put next to each other add up to 10, which makes for easy adding. $4 \times 10 + 5 = 45$. Adding that on to 900 makes 945.

945 is $\boxed{\text{B}}$

Solution 2

We can express each of the terms as a difference from 100 and then add the negatives using $\frac{n(n+1)}{2}$ to get the answer. \begin{align*} (100-10)+(100-9)+\cdots + (100-1) &= 100 \times 10 -(1+2+\cdots +9+10)\\ &= 1000 - 55\\ &= 945 \rightarrow \boxed{\text{B}} \end{align*}

Solution 3

Instead of breaking the sum and then rearranging, we can start by rearranging: \begin{align*} 90+91+92+\cdots +98+99 &=  (90+99)+(91+98)+(92+97)+(93+96)+(94+95) \\ &= 189+189+189+189+189 \\ &= 945\rightarrow \boxed{\text{B}}  \end{align*}

Solution 4

We can use the formula for finite arithmetic sequences.

It is $\frac{n}{2}\times$ ($a_1+a_n$) where $n$ is the number of terms in the sequence, $a_1$ is the first term and $a_n$ is the last term.

Applying it here:

$\frac{10}{2} \times (90+99) = 945 \rightarrow \boxed{B}$

See Also

1985 AJHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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
All AJHSME/AMC 8 Problems and Solutions


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