Difference between revisions of "1964 AHSME Problems/Problem 6"

(Created page with "Since we know the sequence is a geometric sequence, we can set up an equation: <math>\frac{2x+2}{x} = \frac{3x+3}{2x+2}</math>. Solving it, we get: <math>\frac{2x+2}{x} = \...")
 
 
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Since we know the sequence is a geometric sequence, we can set up an equation:
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== Problem ==
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If <math>x, 2x+2, 3x+3, \dots</math> are in geometric progression, the fourth term is:
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<math>\textbf{(A)}\ -27 \qquad
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\textbf{(B)}\ -13\frac{1}{2} \qquad
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\textbf{(C)}\ 12\qquad
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\textbf{(D)}\ 13\frac{1}{2}\qquad
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\textbf{(E)}\ 27 </math> 
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==Solution==
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Since we know the sequence is a geometric sequence, the ratio of consecutive terms is always the same number.  Thus, we can set up an equation:
  
 
<math>\frac{2x+2}{x} = \frac{3x+3}{2x+2}</math>.
 
<math>\frac{2x+2}{x} = \frac{3x+3}{2x+2}</math>.
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<math>\frac{2x+2}{x} = \frac{3x+3}{2x+2}</math>
 
<math>\frac{2x+2}{x} = \frac{3x+3}{2x+2}</math>
  
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<math>(2x+2)(2x+2) = (3x+3)(x)</math>
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<math>4x^2+8x+4 = 3x^2+3x</math>
  
<math>(2x+2)(2x+2) = (3x+3)(x)</math>
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<math>x^2+5x+4 = 0</math>
  
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<math>(x + 4)(x+1) = 0</math>
  
<math>4x^2+8x+4 = 3x^2+3</math>
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<math>x = -4</math> or <math>x = -1</math>
  
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If <math>x=-1</math>, the sequence has a <math>0</math> as the second term, which is not allowed in a geometric sequence, so it is an extraneous solution that came about because we cross-multiplied by <math>(2x + 2)</math>, which is <math>0</math>.
  
<math>x^2+8x+1 = 0</math>.
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If <math>x=-4</math>, we plug into <math>x, 2x +2, 3x + 3</math> to find the sequence starts as <math>-4, -6, -9</math>.  The common ratio is <math>\frac{-6}{-4} = \frac{3}{2}</math>. The next term is <math>\frac{3}{2} \cdot -9 = \frac{-27}{2} = -13\frac{1}{2}</math>, which is option <math>\textbf{(B)}</math>
  
Applying the quadratic formula, we get
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==See Also==
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{{AHSME 40p box|year=1964|num-b=5|num-a=7}}
  
<math>\frac{-b\pm \sqrt{b^2-4ac}}{2a} = \frac{-8\pm \sqrt{8^2-4(1)(1)}}{2(1)} = \frac{-8\pm \sqrt{64-4}}{2} = \frac{-8\pm 2\sqrt{15}}{2} = -4\pm \sqrt{15}</math>.
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[[Category:Introductory Algebra Problems]]
  
From there, you can plug <math>x</math> in to find out the answer.
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{{MAA Notice}}

Latest revision as of 03:12, 23 July 2019

Problem

If $x, 2x+2, 3x+3, \dots$ are in geometric progression, the fourth term is:

$\textbf{(A)}\ -27 \qquad \textbf{(B)}\ -13\frac{1}{2} \qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 13\frac{1}{2}\qquad \textbf{(E)}\ 27$


Solution

Since we know the sequence is a geometric sequence, the ratio of consecutive terms is always the same number. Thus, we can set up an equation:

$\frac{2x+2}{x} = \frac{3x+3}{2x+2}$.

Solving it, we get:

$\frac{2x+2}{x} = \frac{3x+3}{2x+2}$

$(2x+2)(2x+2) = (3x+3)(x)$

$4x^2+8x+4 = 3x^2+3x$

$x^2+5x+4 = 0$

$(x + 4)(x+1) = 0$

$x = -4$ or $x = -1$

If $x=-1$, the sequence has a $0$ as the second term, which is not allowed in a geometric sequence, so it is an extraneous solution that came about because we cross-multiplied by $(2x + 2)$, which is $0$.

If $x=-4$, we plug into $x, 2x +2, 3x + 3$ to find the sequence starts as $-4, -6, -9$. The common ratio is $\frac{-6}{-4} = \frac{3}{2}$. The next term is $\frac{3}{2} \cdot -9 = \frac{-27}{2} = -13\frac{1}{2}$, which is option $\textbf{(B)}$

See Also

1964 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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