1970 AHSME Problems/Problem 22

Problem

If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is

$\text{(A) } 300\quad \text{(B) } 350\quad \text{(C) } 400\quad \text{(D) } 450\quad \text{(E) } 600$

Solution 1

We can setup our first equation as

$\frac{3n(3n+1)}{2} = \frac{n(n+1)}{2} + 150$

Simplifying we get

$9n^2 + 3n = n^2 + n + 300 \Rightarrow 8n^2 + 2n - 300 = 0 \Rightarrow 4n^2 + n - 150 = 0$

So our roots using the quadratic formula are

$\dfrac{-b\pm\sqrt{b^2 - 4ac}}{2a} \Rightarrow \dfrac{-1\pm\sqrt{1^2 - 4\cdot(-150)\cdot4}}{2\cdot4} \Rightarrow \dfrac{-1\pm\sqrt{1+2400}}{8} \Rightarrow 6, -25/4$

Since the question said positive integers, $n = 6$ , so $4n = 24$

$\frac{24\cdot 25}{2} = 300$

$\fbox{A}$

Solution 2

Expressing as an eqaution: $\frac{3n(3n+1)}{2} = \frac{n(n+1)}{2} + 150$ . (1)

The sum of the first 4n positive integers = $\frac{4n(4n+1)}{2}$ . (2)

We will try to rearrange Equation (1) to give equation (2)

$\frac{3n(3n+1)}{2} - \frac{n(n+1)}{2} = 150$ $=\frac{n(3(3n+1)-(n+1))}{2} = 150 =\frac{n(9n+3-n-1)}{2}$ \frac{n(8n+2}{2}= \frac{2n(4n+1}{2} = 150$$ (Error compiling LaTeX. Unknown error_msg)\frac{4n(4n+1}{2} = 2*150 = 300

300 is the answer

1970 AHSC (Problems • Answer Key • Resources)
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1970 AHSME Problems/Problem 22

Contents

Problem

Solution 1

Solution 2

See also