Sequences and Series

The \(n\)th term of a sequence is \(a_{n} = (-1)^{n-1} n^{3}\). What is its 4th term \(a_{4}\)?

1. a\(64\)
2. b\(-64\)
3. c\(-12\)
4. d\(12\)

1. I. A sequence can be regarded as a function whose domain is the set of natural numbers or some subset of it.
2. II. The successive quotients 3, 3.3, 3.33, 3.333, … obtained when 10 is divided by 3 form a finite sequence.
3. III. \(1 + 3 + 5 + 7\) is a finite series with four terms, and the sum of this series is 16.

Which of the statements above about sequences and series is/are correct?

1. aOnly I
2. bOnly I and III
3. cOnly I and II
4. dI, II and III

Match List I (the rule of a sequence) with List II (its first four terms):

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

What is the value of \(\sum_{k=1}^{n} (2 + 3^{k})\)?

1. a\(2n + \dfrac{3^{n} - 1}{2}\)
2. b\(2 + \dfrac{3(3^{n} - 1)}{2}\)
3. c\(2n + \dfrac{3(3^{n} - 1)}{2}\)
4. d\(2n + 3(3^{n} - 1)\)

Assertion (A): The sequence \(0.01, 0.0001, 0.000001, \ldots\) is a geometric progression.

Reason (R): The common ratio of this sequence is \(0.1\).

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

Assertion (A): The sequence \(5, 0, 0, 0, \ldots\) is a G.P. with common ratio \(0\).

Reason (R): A sequence \(a_{1}, a_{2}, a_{3}, \ldots\) is a G.P. if each term is non-zero and \(\dfrac{a_{k+1}}{a_{k}} = r\) (a constant) for \(k \ge 1\).

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 G.P. \(\sqrt{3}, 3, 3\sqrt{3}, \ldots\) is 729?

1. a6th
2. b13th
3. c11th
4. d12th

In a G.P., the 2nd term is 6 and the 5th term is 162. What is its 7th term?

1. a\(1458\)
2. b\(4374\)
3. c\(729\)
4. d\(486\)

Riya sends a safety message to 3 friends: this is the 1st set of messages. Each friend forwards it to 3 people who have not received it (the 2nd set), each of them forwards it to 3 more new people (the 3rd set), and so on. How many people, not counting Riya, have received the message once the 5th set has been sent?

1. a243
2. b364
3. c363
4. d1092

What is the sum to \(n\) terms of the sequence \(4, 44, 444, 4444, \ldots\)?

1. a\(\dfrac{4}{81}(10^{n} - 9n - 1)\)
2. b\(\dfrac{4}{81}(10^{n+1} - 9n - 10)\)
3. c\(\dfrac{4}{9}(10^{n+1} - 9n - 10)\)
4. d\(\dfrac{4}{81}(10^{n+1} + 9n - 10)\)

A G.P. has an even number of terms. The sum of all its terms is 5 times the sum of the terms in the odd places (1st, 3rd, 5th, …). What is its common ratio?

1. a\(\dfrac{1}{4}\)
2. b\(5\)
3. c\(6\)
4. d\(4\)

1. I. If the 4th, 10th and 16th terms of a G.P. are \(x\), \(y\) and \(z\), then \(y^{2} = xz\).
2. II. The products of the corresponding terms of the G.P.s \(a, ar, ar^{2}, \ldots\) and \(A, AR, AR^{2}, \ldots\) form a G.P. with common ratio \(rR\).
3. III. In a G.P. with common ratio 2, the ratio of the sum of the first \(n\) terms to the sum of the terms from the \((n + 1)\)th to the \((2n)\)th is \(2^{n}\).

Which of the statements above is/are correct?

1. aOnly I
2. bOnly II and III
3. cOnly I and II
4. dI, II and III

For the G.P. \(3, 6, 12, 24, \ldots\), match List I with List II:

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

Three numbers are inserted between 2 and 162 so that the five numbers form a G.P. with a positive common ratio. The three numbers are:

1. a\(6, 18, 54\)
2. b\(-6, 18, -54\)
3. c\(42, 82, 122\)
4. d\(10, 18, 90\)

1. I. The geometric mean of 2 and 18 is 10.
2. II. If 6 is the geometric mean of 4 and a positive number \(x\), then \(x = 9\).
3. III. For any two positive numbers \(a\) and \(b\), the numbers \(a, \sqrt{ab}, b\) are consecutive terms of a G.P.

Which of the statements above about the geometric mean is/are correct?

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

Assertion (A): The A.M. of 4 and 25 is greater than their G.M.

Reason (R): For two positive numbers \(a\) and \(b\) with A.M. \(A\) and G.M. \(G\), \(A - G = \dfrac{(\sqrt{a} - \sqrt{b})^{2}}{2}\).

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

The A.M. and the G.M. of the two roots of a quadratic equation are 5 and 4 respectively. Which is the equation?

1. a\(x^{2} + 10x + 16 = 0\)
2. b\(x^{2} - 5x + 4 = 0\)
3. c\(x^{2} - 10x + 4 = 0\)
4. d\(x^{2} - 10x + 16 = 0\)

A school buys a photocopier for AED 10,000. Every year its value falls by 10% of what it was worth at the start of that year. What will the photocopier be worth at the end of 3 years?

1. aAED 7,290
2. bAED 8,100
3. cAED 6,561
4. dAED 7,000

The A.M. and the G.M. of two positive numbers \(a\) and \(b\), where \(a > b\), are in the ratio 5 : 4. What is \(a : b\)?

1. a5 : 4
2. b4 : 1
3. c25 : 16
4. d2 : 1

Let \(a, b, c, d\) and \(p\) be real numbers such that \((a^{2} + b^{2} + c^{2})p^{2} - 2(ab + bc + cd)p + (b^{2} + c^{2} + d^{2}) \le 0\). If \(a = 2\) and \(p = 3\), what is \(d\)?

1. a\(162\)
2. b\(18\)
3. c\(54\)
4. d\(\dfrac{2}{27}\)