Predicting What Comes Next: Sequences and Progressions | Exercise 8.3

Question 4

Which term of the GP: $2, 6, 18, \dots$ is $4374$ ? Write the explicit formula as well as the recursive formula for the $n^{\text{th}}$ term.

Solution

Understand the Question

A Geometric Progression (GP) is a sequence where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio ( $r$ ).
Given GP: $2, 6, 18, \dots$ $2, 6, 18, \dots$
- First term: $a = 2$
- Common ratio: $r = \dfrac{6}{2} = 3$
To solve the problem:
1. Find the explicit formula for the general term: $t_n = a r^{n-1}$ .
2. Equate $t_n = 4374$ and solve for $n$ using powers of $3$ .
3. Write the recursive formula, which defines the first term $t_1$ and gives the rule $t_n = r \cdot t_{n-1}$ for $n \ge 2$ .

Step 1 · Find the Explicit Formula

Diagram 1

First term $a = 2$

Common ratio $r$ :

\begin{aligned} r &= \dfrac{\text{second term}}{\text{first term}} \\[0.6em] &= \dfrac{6}{2} \\[0.6em] &= 3 \end{aligned}

The explicit formula for the $n^{\text{th}}$ term is $t_n = ar^{n-1}$ : $t_n = 2 \times 3^{n-1}$

Step 2 · Find the Term Number for 4374

Set $t_n = 4374$ :

\begin{aligned} 2 \times 3^{n-1} &= 4374 \\[0.6em] 3^{n-1} &= \dfrac{4374}{2} \\[0.6em] 3^{n-1} &= 2187 \end{aligned}

Express $2187$ as a power of $3$ :

\begin{aligned} 3^1 &= 3 \\ 3^2 &= 9 \\ 3^3 &= 27 \\ 3^4 &= 81 \\ 3^5 &= 243 \\ 3^6 &= 729 \\ 3^7 &= 2187 \end{aligned}

Comparing exponents:

\begin{aligned} 3^{n-1} &= 3^7 \\ n - 1 &= 7 \\ n &= 7 + 1 \\ n &= 8 \end{aligned}

Therefore, $4374$ is the $8^{\text{th}}$ term.

Step 3 · Write the Recursive Formula

A recursive formula requires the first term and the relation between consecutive terms:

\begin{aligned} t_1 &= 2 \\ t_n &= 3 \times t_{n-1} \quad \text{for } n \ge 2 \end{aligned}

Answer

(i) $4374$ is the $8^{\text{th}}$ term.

(ii) Explicit formula: $t_n = 2 \times 3^{n-1}$

(iii) Recursive formula: $t_1 = 2$ , $\, t_n = 3t_{n-1} \text{ for } n \ge 2$

Common Mistakes

Exponent Off-by-One Error: Writing the explicit formula as $t_n = ar^n$ instead of $t_n = ar^{n-1}$ , which incorrectly gives $n = 7$ .
Incomplete Recursive Formula: Omitting the base term $t_1 = 2$ or the domain $n \ge 2$ . A recursive definition is incomplete without specifying the starting value.
Power Factorisation Mistake: Making an arithmetic error while expressing $2187$ as $3^7$ .

More questions in Exercise 8.3

Find the $12^{\text{th}}$ term of a GP with common ratio 2, whose $8^{\text{th}}$ term is 192.

Find the $10^{\text{th}}$ and $n^{\text{th}}$ terms of the GP: $5, 25, 125, \dots$ .

A sequence is given by the recursive rule $t_1 = 2$ , $t_{n+1} = 3t_n - 2$ for $n \ge 1$ . Which term of the sequence is 730?

Which term of the GP: $2, 6, 18, \dots$ is $4374$ ? Write the explicit formula as well as the recursive formula for the $n^{\text{th}}$ term.

A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell. It continues bouncing in this way—each time rising to 60% of the previous height.

(i) What height does the ball reach after the $5^{\text{th}}$ bounce? (ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the $6^{\text{th}}$ time?

Which term of the sequence $2, 2\sqrt{2}, 4, \dots$ is 128?

Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on.

Look at Fig. 8.12 and try to answer the following questions.

(i) How many red squares are there in Stages 0 to 3? (ii) Can you predict the number of red squares in Stages 4 and 5? (iii) Can you find a rule for the number of red squares at the $n^{\text{th}}$ stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage. (iv) Suppose the area of the square in Stage 0 is 1 square unit. What is the area of the red region in Stages 1, 2 and 3? What will be the area of the red region in Stages 4 and 5? Find the explicit as well as the recursive formula for the area of the red region at the $n^{\text{th}}$ stage. What happens to this area as $n$ , the number of stages, goes on increasing?