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Sequences and Series: questions with answers

Class 11 Maths, Chapter 8 (NCERT). 15 questions by concept. Open “Show answer” for the answer and why.

How to teach this chapter · Common misconceptions & worked visual for this chapter

A sequence as a function whose domain is the natural numbers

Question 1

How many terms has the sequence 7, 5, 5, 3, 1, 1?

  1. (a)7
  2. (b)4
  3. (c)6
Show answer

(c) 6

A repeated value occupies two positions and is therefore two terms. Counting distinct values would make it a set, which a sequence is not.

Finding which term of a G.P. a given value is

Question 2

For a G.P. with first term 1, finding a term's position gives 26=4n−12^{ 6 } = 4^{n-1}. Which equation in n follows?

  1. (a)6=2n−16 = 2n-1
  2. (b)6=2(n−1)6 = 2(n-1)
  3. (c)6=n−16 = n-1
Show answer

(b) 6=2(n−1)6 = 2(n-1)

Since 4 = 2², the right side becomes 22(n−1)2^{2(n-1)}. Equating exponents now gives 6=2(n−1)6 = 2(n-1).

Finite and infinite sequences

Question 3

The sequence 3, 9, 27, 81, ... is formed by multiplying by 3 each time and never stops. Can a last term be named?

  1. (a)Yes, but only once the whole sequence has been written out.
  2. (b)No: whatever term is chosen, the rule produces another after it.
  3. (c)Yes: the last term is 81.
Show answer

(b) No: whatever term is chosen, the rule produces another after it.

The dots say the rule keeps applying. A term that is printed last is not a last term.

How two given terms fix the whole G.P.

Question 4

In a geometric progression the 3rd term is 54, and the common ratio has been found to be 3. What is the first term?

  1. (a)54
  2. (b)162
  3. (c)6
Show answer

(c) 6

The value given belongs to its own position. Substituting the ratio back is what recovers the first term.

Reading terms off a formula for the nth term

Question 5

A sequence is given by aₙ = n² + 5n + 2. What is a₍5₎?

  1. (a)52
  2. (b)32
  3. (c)37
Show answer

(a) 52

A formula with n in two places needs the position in both. Leaving one standing changes the rule.

The series associated with a sequence

Question 6

How many terms has the series formed from the sequence 3, 6, 12, 24, 48?

  1. (a)4
  2. (b)5
  3. (c)It never ends: a series carries on without a last term.
Show answer

(b) 5

A finite sequence gives a series with exactly as many terms.

What makes a list a sequence, and how its terms are named

Question 7

Two lists are written on the board:

List A: 5, 15, 45

List B: 45, 15, 5

Do the two lists describe the same sequence?

  1. (a)Neither is a sequence until both are written starting from 5.
  2. (b)No: the two lists put different numbers in the first position.
  3. (c)Yes: both lists are made of exactly the same three numbers.
Show answer

(b) No: the two lists put different numbers in the first position.

Order is part of what a sequence IS. The subscript records position, so two lists holding the same values in different places are different sequences.

A sequence given by a recurrence relation

Question 8

A sequence has a₁ = 3 and aₙ = 3aₙ₋₁ + 3 for n > 1. What is a₍4₎?

  1. (a)120
  2. (b)81
  3. (c)15
Show answer

(a) 120

A recurrence has to be worked through from the seed. Substituting the position into the right-hand side answers nothing.

The general term of a G.P.

Question 9

A geometric progression has first term 3 and common ratio 2. What is its 5th term?

  1. (a)96
  2. (b)48
  3. (c)1296
Show answer

(b) 48

Raising the product would multiply by the first term at every step instead of once.

Choosing three unknown terms of a G.P. symmetrically

Question 10

A nonzero three-term G.P. has common ratio r = 4. Allowing a to take a different value in each naming, which namings can represent this G.P.?

  1. (a)Only a ÷ r, a, ar
  2. (b)Both a, ar, ar² and a ÷ r, a, ar
  3. (c)Only a, ar, ar²
Show answer

(b) Both a, ar, ar² and a ÷ r, a, ar

Both namings represent the same ordered triple when a is reassigned. In the second naming, a equals the middle term instead of the first.

Sigma notation for a series

Question 11

How many terms has the sum from k = 3 to 14?

  1. (a)11
  2. (b)17
  3. (c)12
Show answer

(c) 12

The count is upper minus lower PLUS ONE, the same off-by-one that puts the first position at 1 rather than 0.

How many terms of a G.P. reach a given sum

Question 12

A geometric progression has first term 2 and common ratio 2. How many of its terms are needed to reach a total of 62?

  1. (a)4
  2. (b)It cannot be found: no term of the progression has that value.
  3. (c)5
Show answer

(c) 5

The total is the sum, so the sum formula is what gets solved. Looking for a term equal to the total finds nothing, because a sum exceeds every term it is made of.

What makes a progression geometric

Question 13

Is the list 4, 0, 0, 0 a geometric progression?

  1. (a)Yes: its common ratio is 0, since every term after the first is 0.
  2. (b)No: once a term is 0 the next ratio cannot be formed.
  3. (c)Yes, but only if the first term were 1 rather than 4.
Show answer

(b) No: once a term is 0 the next ratio cannot be formed.

The definition requires every term to be non-zero, precisely so that every ratio exists.

The sum of the first n terms of a G.P.

Question 14

A geometric progression has first term 5 and common ratio 2. What is the sum of its first 4 terms?

  1. (a)35
  2. (b)40
  3. (c)75
Show answer

(c) 75

The SUM carries rⁿ. The n − 1 belongs to the general term, which is one term rather than all of them.

The geometric mean of two numbers, and inserting several

Question 15

4 numbers are inserted between two given numbers so that the whole list is a geometric progression. How many terms has that progression altogether?

  1. (a)5
  2. (b)4
  3. (c)6
Show answer

(c) 6

Both ends count. k inserted numbers give k + 2 terms, one more than the number of steps.

Common misconception in Sequences and Series

Open full misconception guide →

Is a term the same as the sum up to it?

Wrong

For 2, 4, 8, the third partial sum is 8.

Why it feels right. I use the third term wherever I see the index 3.

Right

A term is one value. A partial sum adds several values.

Worked example

a₃
8
S₃
2+4+8
14
The third term picks just the third tile. The third partial sum includes all three tiles.

Here a₃ = 8 and S₃ = 2 + 4 + 8 = 14.