Worksheet: Using Arithmetic Sequence Formulas

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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