Worksheet: Using Arithmetic Sequence Formulas

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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,…).

  • A𝑎
  • B𝑎
  • C𝑎
  • D𝑎

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 =29, the value =199
  • Cthe order =27, the value =183
  • Dthe order =2238, the value =190

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 15, and the first term is 25.
  • DThe common difference is 3, and the first term is 55.
  • EThe common difference is 3, and the first term is 1.

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.

  • A𝑎
  • B𝑎
  • C𝑎
  • D𝑎

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,…).

  • A84𝑎+89𝑏
  • B84𝑎+81𝑏
  • C92𝑎+89𝑏
  • D88𝑎+85𝑏

Q14:

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

  • A𝑛=2
  • B𝑛=36
  • C𝑛=9
  • D𝑛=18
  • E𝑛=1

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−1,18
  • B−1,19
  • C1,18
  • D1,19

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.

  • A𝑥=−9,350481, 𝑦=5,502481
  • B𝑥=−14, 𝑦=126
  • C𝑥=−263, 𝑦=23
  • D𝑥=8,434481, 𝑦=−12,282481
  • E𝑥=9,370481, 𝑦=−13,218481

Q18:

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

  • A𝑎=7
  • B𝑎=1
  • C𝑎=9
  • D𝑎=8
  • E𝑎=−1

Q19:

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

  • A𝑑=−1, 𝑎=0
  • B𝑑=−6968, 𝑎=−1134
  • C𝑑=−6967, 𝑎=−4467
  • D𝑑=−6970, 𝑎=1135
  • E𝑑=1, 𝑎=224

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.

Position6789𝑛
Value of Term19222528
  • A𝑛, 22
  • B3𝑛, 54
  • C𝑛+1, 24
  • D3𝑛+1, 55
  • E3𝑛−1, 53

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.

  • A𝑎=−75, 𝑏=−63
  • B𝑎=−74, 𝑏=−64
  • C𝑎=−78, 𝑏=−60
  • D𝑎=−6858, 𝑏=−215
  • E𝑎=−76, 𝑏=−62

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,…,1,000 are not exactly divisible by either 11 or 85.

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