Worksheet: Explicit and Recursive Formulas of Arithmetic Sequences

In this worksheet, we will practice writing explicit and recursive formulas for arithmetic sequences.

Q1:

The third term in an arithmetic sequence is 2 and the sixth term is 11. If the first term is 𝑎 1 , what is an equation for the 𝑛 th term of this sequence?

  • A 𝑎 = 𝑛 1 𝑛
  • B 𝑎 = 7 𝑛 2 𝑛
  • C 𝑎 = 4 𝑛 1 3 𝑛
  • D 𝑎 = 3 𝑛 7 𝑛
  • E 𝑎 = 𝑛 + 5 𝑛

Q2:

The fifth term of an arithmetic sequence is 50 and the tenth term is 25 times greater than the second term. Find the general term 𝑎 𝑛 .

  • A 𝑎 = 5 𝑛 5 𝑛
  • B 𝑎 = 1 5 𝑛 1 0 𝑛
  • C 𝑎 = 1 5 𝑛 4 0 𝑛
  • D 𝑎 = 1 5 𝑛 2 5 𝑛
  • E 𝑎 = 1 5 𝑛 + 5 𝑛

Q3:

Find, in terms of 𝑛 , the general term of an arithmetic sequence whose sixth term is 46 and the sum of the third and tenth term is 102.

  • A 𝑇 = 1 9 4 𝑛 + 3 7 0 𝑛
  • B 𝑇 = 1 4 𝑛 + 1 0 𝑛
  • C 𝑇 = 3 7 0 𝑛 + 1 9 4 𝑛
  • D 𝑇 = 1 0 𝑛 1 4 𝑛
  • E 𝑇 = 1 8 𝑛 + 1 4 𝑛

Q4:

Find, in terms of 𝑛 , the general term of an arithmetic sequence whose ninth term is 7 1 7 and sixteenth term is 1 3 4 7 .

  • A 𝑇 = 2 0 6 4 5 𝑛 + 4 8 4 5 𝑛
  • B 𝑇 = 9 3 𝑛 9 0 𝑛
  • C 𝑇 = 4 8 4 5 𝑛 2 0 6 4 5 𝑛
  • D 𝑇 = 9 0 𝑛 + 9 3 𝑛
  • E 𝑇 = 8 7 𝑛 6 3 0 𝑛

Q5:

Consider the following growing pattern, shown for 𝑛 = 1 , 𝑛 = 2 , and 𝑛 = 3 .

Write an expression for the number of dots in the 𝑛 th such pattern.

  • A 𝑛 + 1
  • B 2 𝑛 + 1
  • C 𝑛 1
  • D 2 𝑛 1
  • E 𝑛 + 2

Q6:

The arithmetic mean between the third and seventh term of a sequence is 36 and the tenth term exceeds the double of the fourth term by 6. Find the general term, 𝑎 , of the arithmetic sequence.

  • A 𝑎 = 𝑛 + 3 1
  • B 𝑎 = 6 𝑛 + 1 2
  • C 𝑎 = 6 𝑛 + 5 4
  • D 𝑎 = 6 𝑛 + 6
  • E 𝑎 = 6 𝑛 + 1 8

Q7:

Find the general term of the arithmetic sequence which satisfies the relations 𝑎 + 𝑎 = 3 0 6 8 and 𝑎 × 𝑎 = 5 2 5 7 9 .

  • A 𝑎 = 5 5 5 2 𝑛 5 7 0 𝑛
  • B 𝑎 = 1 0 𝑛 + 4 5 𝑛
  • C 𝑎 = 1 0 𝑛 2 5 𝑛
  • D 𝑎 = 1 0 𝑛 + 5 5 𝑛
  • E 𝑎 = 1 0 𝑛 + 3 5 𝑛

Q8:

A cable television offers its service at $45 per month and a one-time setup fee of $19.95. Express the total amount paid 𝑃 ( 𝑛 ) after 𝑛 0 months by a recursive formula.

  • A 𝑃 ( 𝑛 + 1 ) = 1 9 . 9 5 + 4 5 𝑛
  • B 𝑃 ( 𝑛 ) = 1 9 . 9 5 + 4 5 𝑛
  • C 𝑃 ( 𝑛 ) = 𝑃 ( 𝑛 1 ) + 4 5 , 𝑃 ( 0 ) = 4 5
  • D 𝑃 ( 𝑛 + 1 ) = 𝑃 ( 𝑛 ) + 4 5 , 𝑃 ( 0 ) = 1 9 . 9 5
  • E 𝑃 ( 𝑛 + 1 ) = 𝑃 ( 𝑛 ) + 4 5 𝑛

Q9:

Find the sequence and its general term of all the even numbers greater than 62.

  • A ( 6 6 , 6 8 , 7 0 , 7 2 , 7 4 , ) , 𝑇 = 2 𝑛 𝑛
  • B ( 6 2 , 6 4 , 6 6 , 6 8 , 7 0 , ) , 𝑇 = 2 𝑛 + 6 2 𝑛
  • C ( 6 4 , 6 6 , 6 8 , 7 0 , 7 2 , ) , 𝑇 = 2 𝑛 𝑛
  • D ( 6 4 , 6 6 , 6 8 , 7 0 , 7 2 , ) , 𝑇 = 2 𝑛 + 6 2 𝑛

Q10:

Find, in terms of 𝑛 , the general term of the sequence 3 5 0 + 1 2 , 3 5 0 + 1 3 , 3 5 0 + 1 4 , 3 5 0 + 1 5 , .

  • A 3 5 0 + ( 1 ) 𝑛 + 1 𝑛
  • B 3 5 0 + ( 1 ) 𝑛 + 1 𝑛 + 1
  • C 3 5 0 + 1 𝑛
  • D 3 5 0 + 1 𝑛 + 1

Q11:

Find, in terms of 𝑛 , the general term of the sequence ( 4 4 , 7 0 , 9 6 , 1 2 2 , ) .

  • A 2 6 𝑛 + 1 8
  • B 2 6 𝑛 1 8
  • C 1 8 𝑛 + 2 6
  • D 2 6 𝑛 + 1 8

Q12:

Engy started working out to get healthier. She worked out for fourteen minutes on the first day and increased her exercise by six minutes every day. Find, in terms of , the term of the sequence which represents her plan.

  • A
  • B
  • C
  • D

Q13:

The graph represents the triangle wave function 𝑇 ( 𝑥 ) , which is periodic, piecewise linear, and defined for all real numbers.

Let 𝑎 𝑛 be the 𝑛 th positive solution to the equation 𝑇 ( 𝑥 ) = 1 . Starting from 𝑎 = 3 2 1 , write a recursive formula for 𝑎 𝑛 .

  • A 𝑎 = 𝑎 + 3 2 𝑛 + 1 𝑛 for 𝑛 1 . 𝑎 = 3 2 1
  • B 𝑎 = 𝑎 + 1 𝑛 + 1 𝑛 for 𝑛 1 . 𝑎 = 3 2 1
  • C 𝑎 = 𝑎 + 1 2 𝑛 + 1 𝑛 for 𝑛 1 . 𝑎 = 3 2 1
  • D 𝑎 = 𝑎 + 2 𝑛 + 1 𝑛 for 𝑛 1 . 𝑎 = 3 2 1
  • E 𝑎 = 𝑎 + 5 2 𝑛 + 1 𝑛 for 𝑛 1 . 𝑎 = 3 2 1

What is the set of numbers which satisfy the equation 𝑇 ( 𝑥 ) = 1 ?

  • A , 7 2 , 3 2 , 1 2 , 5 2 ,
  • B , 7 2 , 3 2 , 3 2 , 7 2 ,
  • C , 5 2 , 1 2 , 1 2 , 5 2 ,
  • D { , 2 , 1 , 1 , 2 , }
  • E , 3 2 , 1 2 , 0 , 1 2 , 5 2 ,

The part of the graph through the origin ( 0 , 0 ) coincides with the line 𝑦 = 2 𝑥 . Use this to find one solution to 𝑇 ( 𝑥 ) = 1 2 . Use the symmetries of the graph to find the next positive solution.

  • A 𝑥 = 1 2 . 𝑥 = 9 4
  • B 𝑥 = 1 2 . 𝑥 = 3 4
  • C 𝑥 = 1 4 . 𝑥 = 3 4
  • D 𝑥 = 3 4 . 𝑥 = 9 4
  • E 𝑥 = 1 4 . 𝑥 = 9 4

Find the first two positive solutions to 𝑇 ( 𝑥 ) = 0 . 3 4 6 .

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

Find the value of 𝑇 ( 𝑒 ) 5 , giving your answer correct to 3 decimal places.

Q14:

The fifth term of an arithmetic sequence is 1 8 7 and the tenth term is 2 times greater than the second term. Find the general term 𝑎 𝑛 .

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

Q15:

The fifth term of an arithmetic sequence is 210 and the tenth term is 3 times greater than the second term. Find the general term 𝑎 𝑛 .

  • A 𝑎 = 2 0 7 5 𝑛 + 6 5 7 5 𝑛
  • B 𝑎 = 3 0 𝑛 + 9 0 𝑛
  • C 𝑎 = 3 0 𝑛 + 3 0 𝑛
  • D 𝑎 = 3 0 𝑛 + 6 0 𝑛
  • E 𝑎 = 3 0 𝑛 + 1 2 0 𝑛

Q16:

Find, in terms of 𝑛 , the general term of an arithmetic sequence whose sixth term is 3 1 and the sum of the third and tenth term is 6 7 .

  • A 𝑇 = 1 2 9 𝑛 2 2 5 𝑛
  • B 𝑇 = 𝑛 5 𝑛
  • C 𝑇 = 2 2 5 𝑛 1 2 9 𝑛
  • D 𝑇 = 5 𝑛 1 𝑛
  • E 𝑇 = 3 𝑛 9 𝑛

Q17:

The fifth term of an arithmetic sequence is 4 and the tenth term is 1 times greater than the second term. Find the general term 𝑎 𝑛 .

  • A 𝑎 = 𝑛 + 2 1 𝑛
  • B 𝑎 = 4 𝑛 + 2 0 𝑛
  • C 𝑎 = 4 𝑛 + 2 8 𝑛
  • D 𝑎 = 4 𝑛 + 2 4 𝑛
  • E 𝑎 = 4 𝑛 + 1 6 𝑛

Q18:

Find, in terms of 𝑛 , the general term of an arithmetic sequence whose sixth term is 30 and the sum of the third and tenth term is 67.

  • A 𝑇 = 1 2 7 𝑛 + 2 4 8 𝑛
  • B 𝑇 = 1 2 𝑛 + 7 𝑛
  • C 𝑇 = 2 4 8 𝑛 + 1 2 7 𝑛
  • D 𝑇 = 7 𝑛 1 2 𝑛
  • E 𝑇 = 5 7 4 𝑛 + 3 7 4 𝑛

Q19:

Find, in terms of 𝑛 , the general term of an arithmetic sequence whose ninth term is 478 and sixteenth term is 891.

  • A 𝑇 = 1 3 6 9 5 𝑛 3 2 1 6 𝑛
  • B 𝑇 = 5 3 𝑛 + 5 9 𝑛
  • C 𝑇 = 3 2 1 6 𝑛 + 1 3 6 9 5 𝑛
  • D 𝑇 = 5 9 𝑛 5 3 𝑛
  • E 𝑇 = 6 5 𝑛 + 4 1 3 𝑛

Q20:

The arithmetic mean between the third and seventh term of a sequence is 9 3 and the tenth term exceeds the double of the fourth term by 44. Find the general term, 𝑎 , of the arithmetic sequence.

  • A 𝑎 = 2 2 3 𝑛 3 8 9 3
  • B 𝑎 = 7 𝑛 6 5
  • C 𝑎 = 7 𝑛 1 1 4
  • D 𝑎 = 7 𝑛 5 8
  • E 𝑎 = 7 𝑛 7 2

Q21:

Find the general term of the arithmetic sequence which satisfies the relations 𝑎 + 𝑎 = 1 4 1 2 1 4 and 𝑎 × 𝑎 = 7 1 3 1 5 .

  • A 𝑎 = 7 2 𝑛 + 1 4 𝑛
  • B 𝑎 = 3 𝑛 + 4 3 𝑛
  • C 𝑎 = 3 𝑛 + 4 𝑛
  • D 𝑎 = 3 𝑛 + 4 6 𝑛
  • E 𝑎 = 3 𝑛 + 4 0 𝑛

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