14. Sequences and Series: problem set

Problem 14.5.1

AMC 8

An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2, 5, 8, 11, 14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$ . Each row and each column in this $5 \times 5$ array is an arithmetic sequence with five terms. The square in the center is labelled $X$ as shown. What is the value of $X$ ?

Problem 14.5.2

Omega Learn Math Contest

There are $4$ positive integers $A$ , $B$ , $C$ , and $D$ . The first $3$ numbers form an arithmetic sequence in that order. The last $3$ numbers form a geometric sequence in that order. The value of $C$ is $3$ times the value of $A$ . What is the smallest possible value of $D$ ?

Problem 14.5.3

AMC 10

Suppose that $\{a_n\}$ is an arithmetic sequence with

a_1 + a_2 + \cdots + a_{100} = 100 \text{ and } a_{101} + a_{102} + \cdots + a_{200} = 200.

What is the value of $a_2 - a_1$ ?

Problem 14.5.4

AMC 10/12

A sequence of three real numbers forms an arithmetic progression with a first term of $9$ . If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?

Problem 14.5.5

AMC 10

In the five-sided star shown, the letters $A$ , $B$ , $C$ , $D$ , and $E$ are replaced by the numbers $3$ , $5$ , $6$ , $7$ , and $9$ , although not necessarily in this order. The sums of the numbers at the ends of the line segments $AB$ , $BC$ , $CD$ , $DE$ , and $EA$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the sequence?

Problem 14.5.6

AIME

The terms of an arithmetic sequence add to $715$ . The first term of the sequence is increased by $1$ , the second term is increased by $3$ , the third term is increased by $5$ , and in general, the $k$ th term is increased by the $k$ th odd positive integer. The terms of the new sequence add to $836$ . Find the sum of the first, last, and middle terms of the original sequence.

Problem 14.5.7(additional)

AHSME

What is the $100$ th number in the arithmetic sequence $1, 5, 9, 13, 17, 21, 25, \ldots$

Problem 14.5.8(additional)

MATHCOUNTS

The non-negative integers $a$ , $b$ , $c$ , $d$ , and $e$ form an arithmetic sequence. If their sum is $440$ , what is the largest possible value for $e$ ?

Problem 14.5.9(additional)

AMC 8

The sum of $25$ consecutive even integers is $10{,}000$ . What is the largest of these $25$ consecutive integers?

Problem 14.5.10(additional)

AMC 10/12

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $100$ cans, how many rows does it contain?

Sequences and Series: all problems