Predicting What Comes Next: Exploring Sequences and Progressions

Which of the following sequences is an arithmetic progression?

1. a\(1, 3, 6, 10, 15, \ldots\)
2. b\(11, 7, 3, -1, -5, \ldots\)
3. c\(1, 4, 9, 16, 25, \ldots\)
4. d\(3, 6, 12, 24, 48, \ldots\)

Using the explicit rule \(u_{n} = 2n - 1\) for the sequence of odd numbers, what is the 53rd odd number?

1. a107
2. b105
3. c106
4. d103

1. I. A sequence is an ordered list of numbers, and each number in it is called a term.
2. II. \(6, 12, 24, 48, 96\) is an infinite sequence.
3. III. The first known written mention of the formula for the sum of the first \(n\) natural numbers is in Āryabhaṭa’s Āryabhaṭīya.

Which of the statements above is/are correct?

1. aOnly I
2. bOnly II and III
3. cI, II and III
4. dOnly I and III

Match List I with List II:

1. a(i)-S, (ii)-Q, (iii)-R, (iv)-P
2. b(i)-Q, (ii)-S, (iii)-R, (iv)-P
3. c(i)-S, (ii)-Q, (iii)-P, (iv)-R
4. d(i)-Q, (ii)-S, (iii)-P, (iv)-R

Assertion (A): The sequence \(1, 2, 3, 5, 8, \ldots\) was first written down explicitly and studied by the Italian mathematician Fibonacci.

Reason (R): Virahānka discovered this sequence in the context of Prakrit meter and poetry.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Which term of the sequence \(t_{n} = 5n - 3\) is 607?

1. a120th
2. b121st
3. c122nd
4. d123rd

A sequence is given by the recursive rule \(u_{1} = 1,\ u_{n} = 2u_{n-1} + 3\) for \(n \ge 2\). Which term of this sequence is 253?

1. a5th
2. b6th
3. c8th
4. d7th

In the Virahānka–Fibonacci sequence \(V_{1} = 1,\ V_{2} = 2,\ V_{n} = V_{n-1} + V_{n-2}\) for \(n \ge 3\), what is \(V_{10}\)?

1. a55
2. b89
3. c144
4. d68

Assertion (A): 471 is not a term of the sequence \(s_{n} = 5n - 2\).

Reason (R): Solving \(5n - 2 = 471\) gives \(n = 94.6\), which is not a natural number.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Match each sequence in List I with the rule for its nth term \(t_{n}\) in List II:

1. a(i)-Q, (ii)-P, (iii)-S, (iv)-R
2. b(i)-R, (ii)-S, (iii)-P, (iv)-Q
3. c(i)-R, (ii)-P, (iii)-S, (iv)-Q
4. d(i)-R, (ii)-P, (iii)-Q, (iv)-S

What is the sum of the first 50 natural numbers, \(1 + 2 + 3 + \ldots + 50\)?

1. a1275
2. b1250
3. c2550
4. d1225

Find the sum \(21 + 22 + 23 + \ldots + 60\).

1. a1599
2. b1830
3. c1620
4. d1640

An auto-rickshaw charges a fixed fee of ₹30 plus ₹15 for every kilometre travelled, so the fares after 1 km, 2 km, 3 km, … form an AP. Meena paid ₹255 for her ride. How many kilometres did she travel?

1. a15 km
2. b17 km
3. c16 km
4. d14 km

The 4th term of an AP is 17 and its 9th term is 37. What is its 20th term?

1. a77
2. b85
3. c97
4. d81

Which term of the GP \(3, 12, 48, \ldots\) is 3072?

1. a5th
2. b6th
3. c8th
4. d7th

Assertion (A): The sequence \(1, -1, 1, -1, 1, \ldots\) is a geometric progression.

Reason (R): In a geometric progression, each term after the first is obtained by adding a fixed number to the previous term.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. The recursive rule \(t_{1} = 3,\ t_{n} = 2t_{n-1}\) for \(n \ge 2\) gives the sequence \(3, 6, 12, 24, \ldots\)
2. II. The explicit rule for the AP \(1, 5, 9, 13, \ldots\) is \(t_{n} = 4n - 3\).
3. III. The common ratio of the GP \(5, 4, \frac{16}{5}, \frac{64}{25}, \ldots\) is \(\frac{4}{5}\).

Which of the statements above about rules for progressions is/are correct?

1. aI, II and III
2. bOnly I and II
3. cOnly II and III
4. dOnly I

1. I. At Stage \(n\) of the Sierpiński triangle there are \(3^{n}\) black triangles.
2. II. If the black region at Stage 0 has area 1 square unit, the black region at Stage 2 has area \(\frac{9}{16}\) square unit.
3. III. As the stage number increases, the area of the black region keeps increasing.

Which of the statements above about the Sierpiński triangle is/are correct?

1. aOnly I
2. bI, II and III
3. cOnly II and III
4. dOnly I and II

How many multiples of 6 lie between 100 and 400?

1. a49
2. b50
3. c66
4. d51

Kabir drops a ball from a height of 32 m. Each time it bounces, it rises to \(\frac{3}{4}\) of the height it fell from. After which bounce does the ball first rise to a height of less than 8 m?

1. a6th bounce
2. b4th bounce
3. c5th bounce
4. d3rd bounce