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Difference between revisions of "1961 AHSME Problems/Problem 26"

(Solution to Problem 26)
 
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Let the first term of the [[arithmetic sequence]] be <math>a</math> and the common difference be <math>d</math>.
 
Let the first term of the [[arithmetic sequence]] be <math>a</math> and the common difference be <math>d</math>.
  
−
The <math>50^{\text{th}}</math> term of the sequence is <math>a+49d</math>, so the sum of the first <math>50</math> terms is <math>\frac{50(a + a + 49d}{2}</math>.
+
The <math>50^{\text{th}}</math> term of the sequence is <math>a+49d</math>, so the sum of the first <math>50</math> terms is <math>\frac{50(a + a + 49d)}{2}</math>.
  
 
The <math>51^{\text{th}}</math> term of the sequence is <math>a+50d</math> and the <math>100^{\text{th}}</math> term of the sequence is <math>a+99d</math>, so the sum of the next <math>50</math> terms is <math>\frac{50(a+50d+a+99d)}{2}</math>.
 
The <math>51^{\text{th}}</math> term of the sequence is <math>a+50d</math> and the <math>100^{\text{th}}</math> term of the sequence is <math>a+99d</math>, so the sum of the next <math>50</math> terms is <math>\frac{50(a+50d+a+99d)}{2}</math>.

Latest revision as of 10:38, 22 May 2018

Problem

For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:

$\textbf{(A)}\ -1221 \qquad \textbf{(B)}\ -21.5 \qquad \textbf{(C)}\ -20.5 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 3.5$

Solution

Let the first term of the arithmetic sequence be $a$ and the common difference be $d$.

The $50^{\text{th}}$ term of the sequence is $a+49d$, so the sum of the first $50$ terms is $\frac{50(a + a + 49d)}{2}$.

The $51^{\text{th}}$ term of the sequence is $a+50d$ and the $100^{\text{th}}$ term of the sequence is $a+99d$, so the sum of the next $50$ terms is $\frac{50(a+50d+a+99d)}{2}$.

Substituting in values results in this system of equations. \[2a+49d=8\] \[2a+149d=108\] Solving for $a$ yields $a = \frac{-41}{2}$. The first term is $-20.5$, which is answer choice $\boxed{\textbf{(C)}}$.

See Also

1961 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 25 		Followed by
Problem 27
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

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