Math Problem Statement

Which of the following is an arithmetic sequence? Options: -20, -25, -30, -35, -45,...; -15.6, -12.9, -10.2, -7.5, -4.8,...; 3.1, 6.2, 12.4, 24.8, 49.6,...; 7.2, 3.6, 1.8, 0.9, 0.45,...

Solution

The question asks, "Which of the following is an arithmetic sequence?" Let's analyze each option:

  1. 20,25,30,35,45,-20, -25, -30, -35, -45, \dots:
    To check if it's an arithmetic sequence, find the difference between consecutive terms:

    • 25(20)=5-25 - (-20) = -5
    • 30(25)=5-30 - (-25) = -5
    • 35(30)=5-35 - (-30) = -5
    • 45(35)=10-45 - (-35) = -10

    The difference is not consistent (changes from 5-5 to 10-10), so it's not an arithmetic sequence.

  2. 15.6,12.9,10.2,7.5,4.8,-15.6, -12.9, -10.2, -7.5, -4.8, \dots:
    Check the differences:

    • 12.9(15.6)=2.7-12.9 - (-15.6) = 2.7
    • 10.2(12.9)=2.7-10.2 - (-12.9) = 2.7
    • 7.5(10.2)=2.7-7.5 - (-10.2) = 2.7
    • 4.8(7.5)=2.7-4.8 - (-7.5) = 2.7

    The difference is consistent, so this is an arithmetic sequence.

  3. 3.1,6.2,12.4,24.8,49.6,3.1, 6.2, 12.4, 24.8, 49.6, \dots:
    Check the ratios (this might suggest a geometric sequence):

    • 6.2/3.1=26.2 / 3.1 = 2
    • 12.4/6.2=212.4 / 6.2 = 2
    • 24.8/12.4=224.8 / 12.4 = 2

    This sequence follows a geometric progression, so it's not an arithmetic sequence.

  4. 7.2,3.6,1.8,0.9,0.45,7.2, 3.6, 1.8, 0.9, 0.45, \dots:
    Check the ratios:

    • 3.6/7.2=0.53.6 / 7.2 = 0.5
    • 1.8/3.6=0.51.8 / 3.6 = 0.5
    • 0.9/1.8=0.50.9 / 1.8 = 0.5

    This is another geometric sequence, so it's not an arithmetic sequence.

Correct Answer:

The arithmetic sequence is 15.6,12.9,10.2,7.5,4.8,-15.6, -12.9, -10.2, -7.5, -4.8, \dots.

Would you like a breakdown of arithmetic sequences or help with another topic?


  1. What is the formula for the nn-th term of an arithmetic sequence?
  2. How can we distinguish between arithmetic and geometric sequences?
  3. What are some real-life applications of arithmetic sequences?
  4. How do you find the sum of the first nn terms of an arithmetic sequence?
  5. Can a sequence be both arithmetic and geometric?

Tip: Always check the consistency of differences between terms to identify an arithmetic sequence.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Sequences
Geometric Sequences

Formulas

Arithmetic sequence formula: a_n = a_1 + (n-1)d

Theorems

Consistent difference for arithmetic sequences

Suitable Grade Level

Grades 6-8