Predicting What Comes Next: Sequences and Progressions | Exercise 8.1

Question 3

Determine whether 97 and 172 are terms of the sequence tn=5n−3t_n = 5n - 3 for n≥1n \ge 1.

Check your answer with HomiSolve it yourself, then let Homi check your steps and spot mistakes.
Solution
Understand the Question
  • A number is a term of the sequence tn=5n−3t_n = 5n - 3 if and only if solving the equation 5n−3=extvalue5n - 3 = ext{value} yields a positive integer nn (i.e., n∈{1,2,3,… }n \in \{1, 2, 3, \dots\}).
  • If nn turns out to be a fraction or not an integer ≥1\ge 1, then the number is not a term of the sequence.

Step 1 · Check whether 97 is a Term

Set tn=97t_n = 97 5n−3=975n - 3 = 97Diagram 1

5n=97+35n=100\begin{aligned} 5n &= 97 + 3 \\ 5n &= 100 \end{aligned}

n=1005=20n = \dfrac{100}{5} = 20

Since n=20n = 20 is a positive integer (n≥1n \ge 1), 9797 is the 20th20^{\text{th}} term of the sequence.

Step 2 · Check whether 172 is a Term

Set tn=172t_n = 172 5n−3=1725n - 3 = 172

5n=172+35n=175\begin{aligned} 5n &= 172 + 3 \\ 5n &= 175 \end{aligned}

n=1755=35n = \dfrac{175}{5} = 35

Since n=35n = 35 is a positive integer (n≥1n \ge 1), 172172 is the 35th35^{\text{th}} term of the sequence.

Answer

Both 9797 and 172172 are terms of the sequence (the 20th20^{\text{th}} and 35th35^{\text{th}} terms, respectively).

Common Mistakes
  • Non-Integer Values of nn: The index nn denotes the position of a term and must strictly be a positive integer (n≥1n \ge 1). If solving for nn results in a fraction, the given number is not a term.
  • Sign Errors: Forgetting to add 33 to both sides before dividing by 55 (e.g., subtracting 33 instead to get 5n=945n = 94).

More questions in Exercise 8.1

Q1

Find the first five terms of the sequence in which the nthn^{\text{th}} term is given by, for n≥1n \ge 1:

(i) tn=3n−4t_n = 3n - 4

(ii) tn=2−5nt_n = 2 - 5n

(iii) tn=n2−2n+3t_n = n^2 - 2n + 3

Q2

Find the 10th10^{\text{th}} and 15th15^{\text{th}} terms of the sequence tn=5n−3t_n = 5n - 3 for n≥1n \ge 1.

Q3

Determine whether 97 and 172 are terms of the sequence tn=5n−3t_n = 5n - 3 for n≥1n \ge 1.

Q4

Which term of the sequence tn=5n−3t_n = 5n - 3 for n≥1n \ge 1 is 607?

Q5

A sequence is given by the recursive rule t1=−5t_1 = -5, tn+1=tn+3t_{n+1} = t_n + 3 for n≥1n \ge 1. Find the first five terms of the sequence. Is 5252 a term of this sequence? If so, which term is it?

Q6

Let T1=1T_1 = 1, T2=2T_2 = 2, T3=4T_3 = 4, and Tn=Tn−1+Tn−2+Tn−3T_n = T_{n-1} + T_{n-2} + T_{n-3} for n≥4n \ge 4. Find T4T_4, T5T_5, T6T_6, T7T_7, and T8T_8.

← Back to Predicting What Comes Next: Sequences and Progressions