Worksheet: Finding the nth Term of an Arithmetic Sequence

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In this worksheet, we will practice writing explicit and recursive formulas for arithmetic sequences to find the value of the nth term in an arithmetic sequence and how to find a term's order given its value.

Q1:

Find the first term of an arithmetic sequence whose common difference is −12, where 𝑎=−540.

Q2:

Find the order of the term whose value is 112 in the sequence (17,22,27,32,…).

  • A 𝑎
  • B 𝑎
  • C 𝑎
  • D 𝑎

Q3:

Find 𝑎, given the arithmetic sequence 𝑎=7𝑛−1.

Q4:

Find the order and value of the first term which is greater than 190 in the arithmetic sequence (−25,−17,−9,…), where 𝑛≥1.

  • Athe order =28, the value =191
  • Bthe order =2238, the value =190
  • Cthe order =27, the value =183
  • Dthe order =29, the value =199

Q5:

Find the tenth term from the end of the arithmetic sequence (−15,−22,−29,…,−260).

Q6:

Find the value of 𝑛 in an arithmetic sequence whose first term is 12, given that 𝑎=187 and 𝑎=369.

Q7:

Find 𝑎, given the arithmetic sequence (18,26,34,…,698), where 𝑛≥1.

Q8:

Find the common difference and first term of an arithmetic sequence given the tenth term is 25 and the fifteenth term is 40.

  • AThe common difference is 3, and the first term is −2.
  • BThe common difference is 13, and the first term is 11.
  • CThe common difference is 3, and the first term is 1.
  • DThe common difference is 15, and the first term is 25.
  • EThe common difference is 3, and the first term is 55.

Q9:

Find the number of terms in the arithmetic sequence (−4,2,8,…,392).

Q10:

Find the number of positive terms in the arithmetic sequence (87,80,73,⋯).

Q11:

Find the order of the term whose value is 115 given the second term of an arithmetic sequence is −20, and the fifth term is 7.

  • A 𝑎
  • B 𝑎
  • C 𝑎
  • D 𝑎

Q12:

Find the order of the term −2 in the arithmetic sequence −16,−13,−12,⋯.

Q13:

Find 𝑎 in the arithmetic sequence (12𝑎+9𝑏,16𝑎+13𝑏,20𝑎+17𝑏,…).

  • A 8 4 𝑎 + 8 9 𝑏
  • B 9 2 𝑎 + 8 9 𝑏
  • C 8 8 𝑎 + 8 5 𝑏
  • D 8 4 𝑎 + 8 1 𝑏

Q14:

Find the value of 𝑛 given 𝑎 from the arithmetic sequence (−46,−37,−28,…) equals 𝑎 from the arithmetic sequence (9,10,11,…).

  • A 𝑛 = 3 6
  • B 𝑛 = 1 8
  • C 𝑛 = 1
  • D 𝑛 = 9
  • E 𝑛 = 2

Q15:

The arithmetic sequence 𝑎 is such that 𝑎−𝑎=70 and 𝑎=5. What is 𝑎?

Q16:

Find 𝑥 and the number of terms in the sequence 4−18𝑥,−30𝑥−7,…,−22𝑥+17,−29𝑥+11.

  • A − 1 , 1 8
  • B 1 , 1 9
  • C 1 , 1 8
  • D − 1 , 1 9

Q17:

Find the values of 𝑥 and 𝑦 given the last term of the arithmetic sequence (−18,𝑥,…,𝑦,130) is 13 times the eighth term.

  • A 𝑥 = 9 , 3 7 0 4 8 1 , 𝑦 = − 1 3 , 2 1 8 4 8 1
  • B 𝑥 = − 1 4 , 𝑦 = 1 2 6
  • C 𝑥 = − 2 6 3 , 𝑦 = 2 3
  • D 𝑥 = 8 , 4 3 4 4 8 1 , 𝑦 = − 1 2 , 2 8 2 4 8 1
  • E 𝑥 = − 9 , 3 5 0 4 8 1 , 𝑦 = 5 , 5 0 2 4 8 1

Q18:

Suppose that 𝑎=5 and 𝑎=3 belong to an arithmetic sequence. What is 𝑎?

  • A 𝑎 = 7
  • B 𝑎 = 1
  • C 𝑎 = 9
  • D 𝑎 = 8
  • E 𝑎 = − 1

Q19:

Find the common difference 𝑑 and the value of 𝑎 given 𝑎=𝑦 and 𝑎=𝑥 where 𝑥=91 and 𝑦=22.

  • A 𝑑 = − 6 9 6 7 , 𝑎 = − 4 4 6 7
  • B 𝑑 = − 6 9 6 8 , 𝑎 = − 1 1 3 4
  • C 𝑑 = − 6 9 7 0 , 𝑎 = 1 1 3 5
  • D 𝑑 = 1 , 𝑎 = 2 2 4
  • E 𝑑 = − 1 , 𝑎 = 0

Q20:

Using the table, determine the expression that represents the value of each term as a function of its position. Then, find the value of the eighteenth term in the sequence.

Position 6 7 8 9 𝑛
Value of Term 19 22 25 28
  • A 3 𝑛 − 1 , 53
  • B 3 𝑛 + 1 , 55
  • C 𝑛 , 22
  • D 𝑛 + 1 , 24
  • E 3 𝑛 , 54

Q21:

Find the order of the term equal to 92 given the sequence 𝑎=7𝑛+71.

Q22:

Find the number of integers between 6 and 138 that are divisible by 4.

Q23:

An arithmetic sequence with 17 terms is (−77,𝑎,…,𝑏,−61). Find the values of 𝑎 and 𝑏.

  • A 𝑎 = − 6 8 5 8 , 𝑏 = − 2 1 5
  • B 𝑎 = − 7 5 , 𝑏 = − 6 3
  • C 𝑎 = − 7 8 , 𝑏 = − 6 0
  • D 𝑎 = − 7 4 , 𝑏 = − 6 4
  • E 𝑎 = − 7 6 , 𝑏 = − 6 2

Q24:

Find the number of negative terms in the arithmetic sequence (−40,−33,−26,⋯).

Q25:

Find how many numbers in the arithmetic sequence 1,2,3,…,1000 are not exactly divisible by either 11 or 85.

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