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Difference between revisions of "2001 CEMC Pascal Problems/Problem 8"

(Created page with "==Problem== In the diagram, the value of <math>x</math> is {{Image needed}} <math> \text{ (A) }\ 100 \qquad\text{ (B) }\ 65 \qquad\text{ (C) }\ 80 \qquad\text{ (D) }\ 70 \qqua...")
 
 
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==Problem==
 
==Problem==
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In the diagram, the value of <math>x</math> is
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The 50th term in the sequence <math>5, 6x, 7x^2, 8x^3, 9x^4, ...</math> is
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{{Image needed}}
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<math> \text{ (A) }\ 100 \qquad\text{ (B) }\ 65 \qquad\text{ (C) }\ 80 \qquad\text{ (D) }\ 70 \qquad\text{ (E) }\ 50</math>
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<math> \text{ (A) }\ 54x^{49} \qquad\text{ (B) }\ 54x^{50} \qquad\text{ (C) }\ 45x^{50} \qquad\text{ (D) }\ 55x^{49} \qquad\text{ (E) }\ 46x^{51}</math>
 
==Solution 1==
 
==Solution 1==
 
We can notice that <math>5 = 5x^{0}</math>, allowing us to see that the next term is the previous term but the coefficient is increased by <math>1</math> each time, and the exponent is increased by <math>1</math>. This means that the coefficient and exponent must have increased <math>49</math> times from the 1st term to the 50th term.
 
We can notice that <math>5 = 5x^{0}</math>, allowing us to see that the next term is the previous term but the coefficient is increased by <math>1</math> each time, and the exponent is increased by <math>1</math>. This means that the coefficient and exponent must have increased <math>49</math> times from the 1st term to the 50th term.

Latest revision as of 10:49, 5 July 2025

Problem

The 50th term in the sequence $5, 6x, 7x^2, 8x^3, 9x^4, ...$ is

$\text{ (A) }\ 54x^{49} \qquad\text{ (B) }\ 54x^{50} \qquad\text{ (C) }\ 45x^{50} \qquad\text{ (D) }\ 55x^{49} \qquad\text{ (E) }\ 46x^{51}$

Solution 1

We can notice that $5 = 5x^{0}$, allowing us to see that the next term is the previous term but the coefficient is increased by $1$ each time, and the exponent is increased by $1$. This means that the coefficient and exponent must have increased $49$ times from the 1st term to the 50th term.

$5 + 49 = 54$, so the coefficient on the 50th term must be $54$.

$0 + 49 = 49$, so the exponent on the 50th term must be $49$.

This corresponds with $\boxed {\textbf {(A) } 54x^{49}}$.

~anabel.disher

Solution 2

We can use the information found in solution 1 to find the expression for the $n$th term of the sequence. Let $a_n$ be the $n$th term of the sequence

Since the coefficient is increased by $1$ after each term, the coefficient must be $n + c$, where $c$ is some number. Using the first term of the sequence, we can see that $1 + c = 5$, so $c = 4$.

Since the exponent is increased by $1$ after each term, the exponent must be $n + d$, where $d$ is some number. Using the first term of the sequence, we can see that $1 + d = 0$, so $d = -1$.

This means that the formula for the $n$th term is:

$a_n = (n + 4) \times x^{n - 1}$

Plugging in $n = 50$, we get:

$a_{50} = (50 + 4) \times x^{50 - 1} = \boxed {\textbf {(A) } 54x^{49}}$

~anabel.disher

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