A sequence is given by the recursive rule , for . Which term of the sequence is 730?

Understand the Question A sequence is defined recursively by and for . Each term is obtained by multiplying the previous term by and subtracting . To determine which term equals , we compute successive terms of the sequence until we reach . Step 1 · Compute Successive Terms of the Sequence Given first term: Using the rule : Since , the required term is the term. Answer > term Common Mistakes Index vs. Term Confusion: Confusing the term number with the term value (e.g., substituting into the formula instead of setting ). Off-by-One Errors: Misidentifying the term number by counting as index rather than . Arithmetic Slip in Multiplication: Making calculation errors in multi-digit multiplications (such as or ) which leads to an incorrect sequence.

Predicting What Comes Next: Sequences and Progressions | Exercise 8.3

Question 3

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?

Solution

Understand the Question

A sequence is defined recursively by $t_1 = 2$ and $t_{n+1} = 3t_n - 2$ for $n \ge 1$ .
Each term is obtained by multiplying the previous term by $3$ and subtracting $2$ .
To determine which term equals $730$ , we compute successive terms of the sequence until we reach $730$ .

Step 1 · Compute Successive Terms of the Sequence

Given first term: $t_1 = 2$

Using the rule $t_{n+1} = 3t_n - 2$ :

\begin{aligned} t_2 &= 3t_1 - 2 \\ &= 3(2) - 2 \\ &= 6 - 2 \\ &= 4 \end{aligned}

\begin{aligned} t_3 &= 3t_2 - 2 \\ &= 3(4) - 2 \\ &= 12 - 2 \\ &= 10 \end{aligned}

\begin{aligned} t_4 &= 3t_3 - 2 \\ &= 3(10) - 2 \\ &= 30 - 2 \\ &= 28 \end{aligned}

\begin{aligned} t_5 &= 3t_4 - 2 \\ &= 3(28) - 2 \\ &= 84 - 2 \\ &= 82 \end{aligned}

\begin{aligned} t_6 &= 3t_5 - 2 \\ &= 3(82) - 2 \\ &= 246 - 2 \\ &= 244 \end{aligned}

\begin{aligned} t_7 &= 3t_6 - 2 \\ &= 3(244) - 2 \\ &= 732 - 2 \\ &= 730 \end{aligned}

Since $t_7 = 730$ , the required term is the $7^{\text{th}}$ term.

Answer

$7^{\text{th}}$ term

Common Mistakes

Index vs. Term Confusion: Confusing the term number $n$ with the term value $t_n$ (e.g., substituting $n=730$ into the formula instead of setting $t_n = 730$ ).
Off-by-One Errors: Misidentifying the term number by counting $t_1$ as index $0$ rather than $1$ .
Arithmetic Slip in Multiplication: Making calculation errors in multi-digit multiplications (such as $3 \times 82$ or $3 \times 244$ ) which leads to an incorrect sequence.

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?