Lesson: Explicit and Recursive Formulas of Arithmetic Sequences

In this lesson, we will learn how to write explicit and recursive formulas for arithmetic sequences.

Sample Question Videos

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  • 03:26
  • 03:17

Worksheet: 21 Questions • 4 Videos

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?

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 .

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.

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 .

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.

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.

Q7:

Find the general term of the arithmetic sequence which satisfies the relations 𝑇 + 𝑇 = 3 0 and 𝑇 × 𝑇 = 5 2 5 .

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.

Q9:

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

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 , .

Q11:

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

Q12:

Isabella 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 𝑛 , the 𝑛 t h term of the sequence which represents her plan.

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 .

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

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.

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

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 .

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 .

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 .

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 .

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.

Q19:

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

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.

Q21:

Find the general term of the arithmetic sequence which satisfies the relations 𝑇 + 𝑇 = 1 4 and 𝑇 × 𝑇 = 7 .

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