Class 11 — Mathematics (NCERT)

Sequences and Series: Arithmetic Progression and Geometric Progression

Class 11

✓By the end of this lesson students will be able to define sequences, series, Arithmetic Progression (AP) and Geometric Progression (GP).
✓By the end of this lesson students will be able to find the n-th term of an AP and a GP.
✓By the end of this lesson students will be able to calculate the sum of the first n terms of an AP and a GP.
✓By the end of this lesson students will be able to understand and apply the concepts of Arithmetic Mean (AM) and Geometric Mean (GM).
✓By the end of this lesson students will be able to solve problems involving properties of AP and GP.

Key concepts

Sequence

A sequence is an ordered list of numbers, where each number is called a term. Sequences can be finite (having a limited number of terms) or infinite (having an unlimited number of terms). For example, 1, 3, 5, 7, ... is an infinite sequence.

Series

A series is the sum of the terms of a sequence. For example, if 1, 3, 5, 7 is a sequence, then 1 + 3 + 5 + 7 is the corresponding series.

Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'. The first term is denoted by 'a'.

n-th term (a_n): a_n = a + (n-1)d

Sum of n terms of an AP (S_n)

The sum of the first 'n' terms of an Arithmetic Progression can be calculated using the following formulas.

S_n = n/2 [2a + (n-1)d] OR S_n = n/2 [a + l], where 'l' is the last term (a_n).

Arithmetic Mean (AM)

If a, A, b are in AP, then A is called the Arithmetic Mean of 'a' and 'b'.

A = (a+b)/2

Geometric Progression (GP)

A Geometric Progression (GP) is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by 'r'. The first term is denoted by 'a'.

n-th term (a_n): a_n = ar^(n-1)

Sum of n terms of a GP (S_n)

The sum of the first 'n' terms of a Geometric Progression can be calculated using the following formulas.

If r = 1, S_n = na. If r ≠ 1, S_n = a(r^n - 1) / (r - 1) OR S_n = a(1 - r^n) / (1 - r).

Geometric Mean (GM)

If a, G, b are in GP, then G is called the Geometric Mean of 'a' and 'b'. For positive numbers 'a' and 'b'.

G = √(ab)

Relationship between AM and GM

For any two positive numbers 'a' and 'b', the Arithmetic Mean is always greater than or equal to the Geometric Mean. Equality holds if and only if a = b.

(a+b)/2 ≥ √(ab)

Key facts to remember

1The n-th term of an AP is a_n = a + (n-1)d.
2The sum of the first n terms of an AP is S_n = n/2 [2a + (n-1)d] or S_n = n/2 [a + l].
3The Arithmetic Mean (AM) of two numbers 'a' and 'b' is (a+b)/2.
4The n-th term of a GP is a_n = ar^(n-1).
5The sum of the first n terms of a GP (r ≠ 1) is S_n = a(r^n - 1) / (r - 1) or S_n = a(1 - r^n) / (1 - r).
6The sum of the first n terms of a GP (r = 1) is S_n = na.
7The Geometric Mean (GM) of two positive numbers 'a' and 'b' is √(ab).
8For any two positive numbers 'a' and 'b', AM ≥ GM.

Worked examples

Example 1

Find the 15th term and the sum of the first 20 terms of the Arithmetic Progression: 3, 7, 11, 15, ...

IGiven AP: 3, 7, 11, 15, ...

IIFirst term, a = 3

IIICommon difference, d = 7 - 3 = 4

IVTo find the 15th term (a_15):

VUsing the formula a_n = a + (n-1)d

VIa_15 = 3 + (15-1)4

VIIa_15 = 3 + (14)4

VIIIa_15 = 3 + 56

9a_15 = 59

10To find the sum of the first 20 terms (S_20):

11Using the formula S_n = n/2 [2a + (n-1)d]

12S_20 = 20/2 [2(3) + (20-1)4]

13S_20 = 10 [6 + (19)4]

14S_20 = 10 [6 + 76]

15S_20 = 10 [82]

16S_20 = 820

Answer

The 15th term is 59. The sum of the first 20 terms is 820.

Always identify 'a' and 'd' correctly before applying the formulas.

Example 2

Find the 7th term and the sum of the first 8 terms of the Geometric Progression: 2, 6, 18, 54, ...

IGiven GP: 2, 6, 18, 54, ...

IIFirst term, a = 2

IIICommon ratio, r = 6/2 = 3

IVTo find the 7th term (a_7):

VUsing the formula a_n = ar^(n-1)

VIa_7 = 2 * (3)^(7-1)

VIIa_7 = 2 * (3)^6

VIIIa_7 = 2 * 729

9a_7 = 1458

10To find the sum of the first 8 terms (S_8):

11Since r = 3 ≠ 1, use S_n = a(r^n - 1) / (r - 1)

12S_8 = 2((3)^8 - 1) / (3 - 1)

13S_8 = 2(6561 - 1) / 2

14S_8 = 6560

Answer

The 7th term is 1458. The sum of the first 8 terms is 6560.

Be careful with powers of the common ratio 'r'.

Example 3

The sum of the first three terms of a GP is 39/10 and their product is 1. Find the common ratio and the terms of the GP.

ILet the three terms of the GP be a/r, a, ar.

IIGiven that their product is 1:

III(a/r) * a * (ar) = 1

IVa^3 = 1

VSince 'a' is a real number, a = 1.

VINow, the terms are 1/r, 1, r.

VIIGiven that the sum of the first three terms is 39/10:

VIII1/r + 1 + r = 39/10

9Multiply by 10r to clear denominators (assuming r ≠ 0):

1010 + 10r + 10r^2 = 39r

11Rearrange into a quadratic equation:

1210r^2 - 29r + 10 = 0

13Factorise the quadratic equation:

1410r^2 - 25r - 4r + 10 = 0

155r(2r - 5) - 2(2r - 5) = 0

16(5r - 2)(2r - 5) = 0

17This gives two possible values for r:

185r - 2 = 0 ⇒ r = 2/5

192r - 5 = 0 ⇒ r = 5/2

20Case 1: If r = 2/5

21The terms are 1/(2/5), 1, 2/5 i.e., 5/2, 1, 2/5.

22Case 2: If r = 5/2

23The terms are 1/(5/2), 1, 5/2 i.e., 2/5, 1, 5/2.

Answer

The common ratio is 2/5 or 5/2. The terms of the GP are (5/2, 1, 2/5) or (2/5, 1, 5/2).

When dealing with a product of terms in GP, assuming terms as a/r, a, ar simplifies calculations significantly.

Common mistakes

✗Confusing the formulas for AP and GP, especially for the n-th term and sum of n terms.
✗Incorrectly calculating the common difference 'd' or common ratio 'r', particularly with negative numbers or fractions.
✗Forgetting the special case for the sum of a GP when the common ratio 'r' is equal to 1.
✗Algebraic errors when solving equations involving AP or GP, such as sign errors or calculation mistakes.
✗Not checking if the given sequence is an AP or GP before applying the respective formulas.

Exam tips

★Carefully read the question to determine whether it involves an AP or a GP, and whether you need to find a specific term or a sum.
★Always write down the values of 'a' (first term), 'd' (common difference) or 'r' (common ratio), and 'n' (number of terms) clearly before starting calculations.
★Memorise all the formulas for AP and GP (n-th term, sum of n terms, AM, GM) to save time and ensure accuracy.
★Show all steps of your working clearly and logically, as this helps in gaining partial marks even if the final answer is incorrect.
★Practice a variety of problems, including those that combine concepts or require solving quadratic equations, to build confidence and speed.

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