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Lesson: Finding the Arithmetic Sequence

In this lesson, we will learn how to find arithmetic sequences by using arithmetic sequence general term formula and techniques like solving a system of two linear equations simultaneously.

Worksheet • 25 Questions

Q1:

Find the arithmetic sequence in which π‘Ž + π‘Ž = βˆ’ 2 8 2 4 and π‘Ž Γ— π‘Ž = 1 4 0 3 5 .

  • A ( βˆ’ 1 8 , βˆ’ 1 6 , βˆ’ 1 4 , … )
  • B ( βˆ’ 1 6 , βˆ’ 1 4 , βˆ’ 1 2 , … )
  • C ( βˆ’ 3 4 , βˆ’ 3 2 , βˆ’ 3 0 , … )
  • D ( βˆ’ 3 6 , βˆ’ 3 4 , βˆ’ 3 2 , … )

Q2:

Find the arithmetic sequence in which π‘Ž + π‘Ž = βˆ’ 2 2 4 and π‘Ž Γ— π‘Ž = βˆ’ 1 7 3 5 .

  • A ( βˆ’ 1 9 , βˆ’ 1 0 , βˆ’ 1 , … )
  • B ( βˆ’ 1 0 , βˆ’ 1 , 8 , … )
  • C ( βˆ’ 2 9 , βˆ’ 2 0 , βˆ’ 1 1 , … )
  • D ( βˆ’ 3 8 , βˆ’ 2 9 , βˆ’ 2 0 , … )

Q3:

Find the 7-term arithmetic sequence whose middle term is βˆ’ 1 0 5 and sum of the last 3 terms is βˆ’ 3 6 9 .

  • A βˆ’ 7 8 , βˆ’ 8 7 , βˆ’ 9 6 , … , βˆ’ 1 3 2
  • B 2 7 , βˆ’ 1 7 , βˆ’ 6 1 , … , βˆ’ 1 3 2
  • C βˆ’ 7 8 , βˆ’ 6 9 , βˆ’ 6 0 , … , βˆ’ 1 3 2
  • D βˆ’ 8 7 , βˆ’ 9 3 , βˆ’ 9 9 , … , βˆ’ 1 3 2
  • E βˆ’ 9 , βˆ’ 8 7 , βˆ’ 1 6 5 , … , βˆ’ 1 3 2

Q4:

Find the 59-term arithmetic sequence whose middle term is 133 and sum of the last 3 terms is 735.

  • A 1 7 , 2 1 , 2 5 , … , 2 4 9
  • B βˆ’ 4 4 9 6 , βˆ’ 2 0 3 3 , βˆ’ 1 2 1 2 , … , 2 4 9
  • C 1 7 , 1 3 , 9 , … , 2 4 9
  • D 2 1 , 7 2 1 2 9 , 8 3 3 2 9 , … , 2 4 9
  • E 4 , 2 1 , 3 8 , … , 2 4 9

Q5:

Find the four numbers which form an arithmetic sequence given that the sum of the four terms equals 362, and the sum of the squares of the first and fourth terms exceeds the sum of the squares of the second and third terms by 900.

  • A 6 8 , 8 3 , 9 8 , 1 1 3
  • B βˆ’ 7 1 9 2 , βˆ’ 2 6 9 2 , 1 8 1 2 , 6 3 1 2
  • C 6 8 , 5 3 , 3 8 , 2 3
  • D 2 4 1 2 , 2 1 1 2 , 1 8 1 2 , 1 5 1 2
  • E 1 5 , 8 3 , 1 5 1 , 2 1 9

Q6:

Find the four numbers which form an arithmetic sequence given that the sum of the four terms equals βˆ’ 9 2 , and the sum of the squares of the first and fourth terms exceeds the sum of the squares of the second and third terms by 576.

  • A βˆ’ 4 1 , βˆ’ 2 9 , βˆ’ 1 7 , βˆ’ 5
  • B βˆ’ 3 1 1 , βˆ’ 1 6 7 , βˆ’ 2 3 , 1 2 1
  • C βˆ’ 4 1 , βˆ’ 5 3 , βˆ’ 6 5 , βˆ’ 7 7
  • D 1 , βˆ’ 1 1 , βˆ’ 2 3 , βˆ’ 3 5
  • E 1 2 , βˆ’ 2 9 , βˆ’ 7 0 , βˆ’ 1 1 1

Q7:

Find the five numbers that make an increasing arithmetic sequence, given their sum is βˆ’ 5 0 and the squares of the smallest and largest terms add up to 1 000.

  • A βˆ’ 3 0 , βˆ’ 2 0 , βˆ’ 1 0 , 0 , 1 0
  • B βˆ’ 2 1 0 , βˆ’ 1 1 0 , βˆ’ 1 0 , 9 0 , 1 9 0
  • C βˆ’ 3 0 , βˆ’ 4 0 , βˆ’ 5 0 , βˆ’ 6 0 , βˆ’ 7 0
  • D 1 0 , 0 , βˆ’ 1 0 , βˆ’ 2 0 , βˆ’ 3 0
  • E 1 0 , βˆ’ 2 0 , βˆ’ 5 0 , βˆ’ 8 0 , βˆ’ 1 1 0

Q8:

Find the five numbers that make an increasing arithmetic sequence, given their sum is βˆ’ 4 5 and the squares of the smallest and largest terms add up to 450.

  • A 3 , βˆ’ 3 , βˆ’ 9 , βˆ’ 1 5 , βˆ’ 2 1
  • B βˆ’ 8 1 , βˆ’ 4 5 , βˆ’ 9 , 2 7 , 6 3
  • C 3 , 9 , 1 5 , 2 1 , 2 7
  • D βˆ’ 2 1 , βˆ’ 1 5 , βˆ’ 9 , βˆ’ 3 , 3
  • E βˆ’ 6 , βˆ’ 3 , 0 , 3 , 6

Q9:

Find the three consecutive terms of an arithmetic sequence given their sum is 243, and if βˆ’ 3 6 is added to their second term, the multiplicative inverse of the three terms will form another arithmetic sequence.

  • A 1 3 5 , 8 1 , 2 7
  • B 1 1 3 5 , 1 8 1 , 1 2 7
  • C 2 7 , βˆ’ 2 7 , βˆ’ 8 1
  • D 1 1 3 5 , 1 4 5 , 1 2 7
  • E 1 3 5 , 1 8 9 , 2 4 3

Q10:

Find the three consecutive terms of an arithmetic sequence given their sum is βˆ’ 8 1 0 , and if 30 is added to their second term, the multiplicative inverse of the three terms will form another arithmetic sequence.

  • A βˆ’ 1 8 0 , βˆ’ 2 7 0 , βˆ’ 3 6 0
  • B βˆ’ 1 1 8 0 , βˆ’ 1 2 7 0 , βˆ’ 1 3 6 0
  • C βˆ’ 3 6 0 , βˆ’ 4 5 0 , βˆ’ 5 4 0
  • D βˆ’ 1 1 8 0 , βˆ’ 1 2 4 0 , βˆ’ 1 3 6 0
  • E βˆ’ 1 8 0 , βˆ’ 9 0 , 0

Q11:

Find the arithmetic sequence given , and all terms are positive.

  • A
  • B
  • C
  • D

Q12:

Find the arithmetic sequence given , and all terms are positive.

  • A
  • B
  • C
  • D

Q13:

Find the arithmetic sequence whose ninth term is βˆ’ 1 1 9 , and the arithmetic mean between the third and fifth terms is βˆ’ 6 9 .

  • A ( βˆ’ 3 9 , βˆ’ 4 9 , βˆ’ 5 9 , … )
  • B ο€Ό βˆ’ 1 9 , βˆ’ 6 3 2 , βˆ’ 4 4 , … 
  • C ( βˆ’ 1 0 , βˆ’ 4 9 , βˆ’ 8 8 , … )
  • D ( βˆ’ 3 9 , βˆ’ 2 9 , βˆ’ 1 9 , … )
  • E ( βˆ’ 4 9 , βˆ’ 5 9 , βˆ’ 6 9 , … )

Q14:

Find the arithmetic sequence whose ninth term is βˆ’ 5 9 , and the arithmetic mean between the third and fifth terms is βˆ’ 7 4 .

  • A ( βˆ’ 8 3 , βˆ’ 8 0 , βˆ’ 7 7 , … )
  • B ο€Ό βˆ’ 8 9 , βˆ’ 3 4 1 4 , βˆ’ 1 6 3 2 , β‹― 
  • C ( 3 , βˆ’ 8 0 , βˆ’ 1 6 3 , … )
  • D ( βˆ’ 8 3 , βˆ’ 8 6 , βˆ’ 8 9 , … )
  • E ( βˆ’ 8 0 , βˆ’ 7 7 , βˆ’ 7 4 , … )

Q15:

Find the four numbers which form an arithmetic sequence given that the sum of the four terms equals βˆ’ 1 1 6 and the sum of the squares of the four terms equals 3 444.

  • A βˆ’ 3 5 , βˆ’ 3 1 , βˆ’ 2 7 , βˆ’ 2 3
  • B βˆ’ 6 1 , βˆ’ 4 5 , βˆ’ 2 9 , βˆ’ 1 3
  • C βˆ’ 3 5 , βˆ’ 3 9 , βˆ’ 4 3 , βˆ’ 4 7
  • D βˆ’ 2 1 , βˆ’ 2 5 , βˆ’ 2 9 , βˆ’ 3 3
  • E 4 , βˆ’ 3 1 , βˆ’ 6 6 , βˆ’ 1 0 1

Q16:

Find the four numbers which form an arithmetic sequence given that the sum of the four terms equals βˆ’ 3 2 and the sum of the squares of the four terms equals 2 676.

  • A βˆ’ 4 1 , βˆ’ 1 9 , 3 , 2 5
  • B βˆ’ 9 7 6 , βˆ’ 4 9 2 , βˆ’ 8 , 4 7 6
  • C βˆ’ 4 1 , βˆ’ 6 3 , βˆ’ 8 5 , βˆ’ 1 0 7
  • D 3 6 , 1 4 , βˆ’ 8 , βˆ’ 3 0
  • E 2 2 , βˆ’ 1 9 , βˆ’ 6 0 , βˆ’ 1 0 1

Q17:

Find the arithmetic sequence for which and is the additive inverse of .

  • A
  • B
  • C
  • D
  • E

Q18:

Find the arithmetic sequence for which and is the additive inverse of .

  • A
  • B
  • C
  • D
  • E

Q19:

Find an arithmetic sequence given and .

  • A
  • B
  • C
  • D
  • E

Q20:

Find an arithmetic sequence given and .

  • A
  • B
  • C
  • D
  • E

Q21:

Find the arithmetic sequence in which the sum of the first and third terms equals βˆ’ 1 4 2 , and the sum of its third and fourth terms equals βˆ’ 1 5 1 .

  • A ( βˆ’ 6 8 , βˆ’ 7 1 , βˆ’ 7 4 , β‹― )
  • B ( βˆ’ 7 1 , βˆ’ 7 4 , βˆ’ 7 7 , β‹― )
  • C ( βˆ’ 1 3 9 , βˆ’ 1 4 2 , βˆ’ 1 4 5 , β‹― )
  • D ( βˆ’ 1 3 6 , βˆ’ 1 3 9 , βˆ’ 1 4 2 , β‹― )

Q22:

The first and last terms of an arithmetic sequence are βˆ’ 5 5 and 209 respectively. There are 21 terms between the first and last terms. Find the a list of these intermediate terms.

  • A ( βˆ’ 4 3 , βˆ’ 3 1 , βˆ’ 1 9 , … , 1 9 7 )
  • B ( βˆ’ 4 8 , βˆ’ 4 1 , βˆ’ 3 4 , … , 9 2 )
  • C ( βˆ’ 5 5 , βˆ’ 4 3 , βˆ’ 3 1 , … , 2 0 9 )
  • D ( βˆ’ 4 4 , βˆ’ 3 3 , βˆ’ 2 2 , … , 1 7 6 )
  • E ( βˆ’ 3 1 , βˆ’ 7 , 1 7 , … , 1 7 3 )

Q23:

Find the three numbers which form an arithmetic sequence given the sum is 69 and product is 6 279.

  • A 7 , 2 3 , 3 9
  • B βˆ’ 4 8 9 , βˆ’ 2 3 3 , 2 3
  • C 7 , βˆ’ 9 , βˆ’ 2 5
  • D 5 5 , 3 9 , 2 3
  • E 1 6 , 2 3 , 3 0

Q24:

What is the next number in the sequence √ 6 , √ 2 4 , √ 5 4 , √ 9 6 ?

  • A √ 1 5 0
  • B √ 1 3 8
  • C √ 1 8 0
  • D √ 1 2 0

Q25:

Find the arithmetic sequence whose twentieth term is 28, given that the sum of its third and sixth terms is greater than its ninth term by 8.

  • A ( 9 , 1 0 , 1 1 , β‹― )
  • B ( 1 0 , 1 1 , 1 2 , β‹― )
  • C ( 1 9 , 2 0 , 2 1 , β‹― )
  • D ( 1 8 , 1 9 , 2 0 , β‹― )
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