Predicting What Comes Next: Sequences and Progressions

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

(i) $t_n = 3n - 4$

(ii) $t_n = 2 - 5n$

(iii) $t_n = n^2 - 2n + 3$

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

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

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

A sequence is given by the recursive rule $t_1 = -5$, $t_{n+1} = t_n + 3$ for $n \ge 1$. Find the first five terms of the sequence. Is 52 a term of this sequence? If so, which term is it?

Let $T_1 = 1$, $T_2 = 2$, $T_3 = 4$, and $T_n = T_{n-1} + T_{n-2} + T_{n-3}$ for $n \ge 4$. Find $T_4, T_5, T_6, T_7,$ and $T_8$.

Find the $10^{\text{th}}$ and $26^{\text{th}}$ terms of the AP: 3, 8, 13, 18, ....

Which term of the AP : 21, 18, 15, ... is $-81$? Also, is 0 a term of this AP? Give reasons for your answer.

Find the $n^{\text{th}}$ term of the AP: 11, 8, 5, 2 ... Write the recursive rule for this AP.

An AP consists of 50 terms in which the $3^{\text{rd}}$ term is 12 and the last term is 106. Find the $29^{\text{th}}$ term.

(Hint: If 'a' is the first term and 'd' the common difference, then we arrive at the equations $a + 2d = 12$ and $a + 49d = 106$. Solve this pair of linear equations for 'a' and 'd'.)

How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?

Harish started work at an annual salary of ₹5,00,000 and received an increment of ₹20,000 each year. After how many years did his income reach ₹7,00,000?

A child arranges marbles in rows so that the first row has 1 marble, the second has 2 marbles, the third has 3, and so on up to 25 rows. How many marbles does the child use in all?

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, ... .

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, ... 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?

Find the $31^{\text{st}}$ term of an AP whose $11^{\text{th}}$ term is 38 and $16^{\text{th}}$ term is 73.

Determine the AP whose third term is 16 and whose $7^{\text{th}}$ term exceeds the $5^{\text{th}}$ term by 12.

How many three-digit numbers are divisible by 7? (Hint: All three-digit numbers divisible by 7 form an AP. Find the smallest and largest such three-digit numbers.)

How many multiples of 4 lie between 10 and 250? (Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)

Find a GP for which the sum of the first two terms is $-4$ and the fifth term is 4 times the third term.

Find all possible ways of expressing 100 as the sum of consecutive natural numbers.

The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the $2^{\text{nd}}$ hour, $4^{\text{th}}$ hour and $n^{\text{th}}$ hour?

The sum of the $4^{\text{th}}$ and $8^{\text{th}}$ terms of an AP is 24 and the sum of the $6^{\text{th}}$ and $10^{\text{th}}$ terms is 44. Find the first three terms of the AP.

Find the smallest value of $n$ such that the sum of the first $n$ natural numbers is greater than 1,000.

Which term of the GP: 2, 8, 32, ... is 131072? Write the explicit formula as well as the recursive formula for the $n^{\text{th}}$ term.

The sum of the first three terms of a GP is $\frac{13}{12}$ and their product is $-1$. Find the common ratio and the terms.

If the $4^{\text{th}}$, $10^{\text{th}}$ and $16^{\text{th}}$ terms of a GP are $x$, $y$ and $z$ respectively, prove that $x, y, z$ are in GP.

The sum of the first three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.

Suppose $P_1 = 1$, $P_2 = 2$ and for $n > 2$, $P_n = P_1 + P_2 + \dots + P_{n-1} + 1$. Find the values of $P_1, P_2, \dots, P_8$. Can you find a simpler recursive formula for $P_n$? Can you give an explicit formula?

Suppose $W_1 = 1$, $W_2 = 2$ and for $n > 2$, $W_n = W_1 + W_2 + \dots + W_{n-2} + 2$. Find the values of $W_1, W_2, \dots, W_8$. Do you recognise this sequence?