2001 CEMC Pascal Problems/Problem 8

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 $d = -1$ .