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AMC 10/12

Arithmetic Sequences: all problems

Focus mode: every practice problem from the chapter, one after another.

0 / 12 solved

Problem 15.1.1

AMC 10

What is the greatest number of consecutive integers whose sum is 4545?

Problem 15.1.2

AMC 10

In how many ways can 345345 be written as the sum of an increasing sequence of two or more consecutive positive integers?

Problem 15.1.3

AMC 10/12

Suppose that SS is a finite set of positive integers. If the greatest integer in SS is removed from SS, then the average value (arithmetic mean) of the integers remaining is 3232. If the least integer in SS is also removed, then the average value of the integers remaining is 3535. If the greatest integer is then returned to the set, the average value of the integers rises to 4040. The greatest integer in the original set SS is 7272 greater than the least integer in SS. What is the average value of all the integers in the set SS?

Problem 15.1.4

AMC 10

In the five-sided star shown, the letters AA, BB, CC, DD, and EE are replaced by the numbers 33, 55, 66, 77, and 99, although not necessarily in this order. The sums of the numbers at the ends of the line segments AB‾\overline{AB}, BC‾\overline{BC}, CD‾\overline{CD}, DE‾\overline{DE}, and EA‾\overline{EA} form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?

Problem 15.1.5

AMC 10

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 100100 cans, how many rows does it contain?

4 terms from 1 with difference 2: a_4 = 1 + (4 − 1) × 2 = 7; sum = 16

first term a₁1
difference d2
terms n4
look at term k4
The terms on a number line
1+23+25+27
The same terms as a staircase
1357average 4
a4 = 1 + (4 − 1) × 2 = 7
average = (1 + 7) ÷ 2 = 4; sum = 4 × 4 = 16

a_4 = 1 + (4 − 1) × 2 = 7; average 4, sum 16

Problem 15.1.6

AIME

The terms of an arithmetic sequence add to 715715. The first term of the sequence is increased by 11, the second term is increased by 33, the third term is increased by 55, and in general, the kkth term is increased by the kkth odd positive integer. The terms of the new sequence add to 836836. Find the sum of the first, last, and middle terms of the original sequence.

Problem 15.1.7

AMC 10

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

a1+a2+⋯+a100=100 and a101+a102+⋯+a200=200.a_1 + a_2 + \cdots + a_{100} = 100 \text{ and } a_{101} + a_{102} + \cdots + a_{200} = 200.

What is the value of a2−a1a_2 - a_1?

Problem 15.1.8(additional)

AMC 10

Mary divides a circle into 1212 sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

Problem 15.1.9(additional)

AMC 10

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

Problem 15.1.10(additional)

AIME

The degree measures of the angles in a convex 1818-sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

Problem 15.1.11(additional)

Find the roots of the polynomial x5−5x4−35x3+ax2+bx+cx^5 - 5x^4 - 35x^3 + ax^2 + bx + c, given that the roots form an arithmetic progression.

Problem 15.1.12(additional)

AIMEChallenge

Find the eighth term of the sequence 1440,1716,1848,…1440, 1716, 1848, \ldots, whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.