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Class 10 — Mathematics (NCERT)

Arithmetic Progressions

Class 10

  • ✓By the end of this lesson students will be able to define an Arithmetic Progression (A.P.) and identify its common difference.
  • ✓By the end of this lesson students will be able to derive and apply the formula for the nth term of an A.P.
  • ✓By the end of this lesson students will be able to derive and apply the formula for the sum of the first n terms of an A.P.
  • ✓By the end of this lesson students will be able to solve real-life problems involving A.P.s.

Key concepts

Arithmetic Progression (A.P.)

An Arithmetic Progression (A.P.) is a sequence of numbers such that the difference between any term and its preceding term is constant. This constant difference is called the common difference.

Common Difference (d)

The constant difference between any term and its preceding term in an A.P. It is denoted by 'd'. If a sequence is a_1, a_2, a_3, ..., then d = a_2 - a_1 = a_3 - a_2, and so on.

d = a_n - a_{n-1}
General Form of an A.P.

If 'a' is the first term and 'd' is the common difference, then an A.P. can be written as:

a, a+d, a+2d, a+3d, ...
nth Term of an A.P. (a_n)

The formula to find any term in an A.P. without listing all the terms. Here, 'a' is the first term, 'd' is the common difference, and 'n' is the term number.

a_n = a + (n-1)d
Sum of the First n Terms of an A.P. (S_n)

The sum of the first 'n' terms of an A.P. Here, 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.

S_n = n/2 [2a + (n-1)d]
Alternative Formula for Sum of n Terms

If the first term 'a' and the last term 'l' (or a_n) of an A.P. are known, the sum of the first 'n' terms can also be found using this formula.

S_n = n/2 [a + l]

Key facts to remember

  • 1An A.P. is a sequence where the difference between consecutive terms is constant, known as the common difference 'd'.
  • 2The general form of an A.P. is a, a+d, a+2d, a+3d, ... where 'a' is the first term.
  • 3The formula for the nth term of an A.P. is a_n = a + (n-1)d.
  • 4The formula for the sum of the first n terms of an A.P. is S_n = n/2 [2a + (n-1)d].
  • 5If the last term 'l' (or a_n) is known, the sum of the first n terms can also be calculated as S_n = n/2 [a + l].
  • 6If a, b, c are in A.P., then 2b = a + c (b is the arithmetic mean of a and c).

Worked examples

Example 1

Find the 15th term of the A.P.: 3, 7, 11, 15, ...

IGiven A.P. is 3, 7, 11, 15, ...
IIHere, the first term, a = 3.
IIIThe common difference, d = 7 - 3 = 4.
IVWe need to find the 15th term, so n = 15.
VUsing the formula for the nth term of an A.P., a_n = a + (n-1)d.
VISubstitute the values: a_15 = 3 + (15-1) × 4.
VIIa_15 = 3 + 14 × 4.
VIIIa_15 = 3 + 56.
9a_15 = 59.

Answer

The 15th term of the A.P. is 59.

Always identify 'a', 'd', and 'n' clearly before applying the formula.

Example 2

Which term of the A.P.: 21, 18, 15, ... is -81?

IGiven A.P. is 21, 18, 15, ...
IIHere, the first term, a = 21.
IIIThe common difference, d = 18 - 21 = -3.
IVLet the nth term be -81, so a_n = -81.
VUsing the formula for the nth term of an A.P., a_n = a + (n-1)d.
VISubstitute the values: -81 = 21 + (n-1)(-3).
VII-81 - 21 = (n-1)(-3).
VIII-102 = -3(n-1).
9Divide both sides by -3: (-102) / (-3) = n-1.
1034 = n-1.
11n = 34 + 1.
12n = 35.

Answer

The 35th term of the A.P. is -81.

Be careful with negative signs when calculating the common difference and during algebraic manipulation.

Example 3

Find the sum of the first 22 terms of the A.P.: 8, 3, -2, ...

IGiven A.P. is 8, 3, -2, ...
IIHere, the first term, a = 8.
IIIThe common difference, d = 3 - 8 = -5.
IVWe need to find the sum of the first 22 terms, so n = 22.
VUsing the formula for the sum of the first n terms of an A.P., S_n = n/2 [2a + (n-1)d].
VISubstitute the values: S_22 = 22/2 [2(8) + (22-1)(-5)].
VIIS_22 = 11 [16 + (21)(-5)].
VIIIS_22 = 11 [16 - 105].
9S_22 = 11 [-89].
10S_22 = -979.

Answer

The sum of the first 22 terms of the A.P. is -979.

Ensure correct order of operations (BODMAS) when simplifying expressions.

Common mistakes

  • ✗Confusing 'n' (number of terms) with 'a_n' (the nth term itself).
  • ✗Incorrectly calculating the common difference 'd', especially when terms are decreasing or negative.
  • ✗Errors in algebraic manipulation when solving for 'a', 'd', or 'n' from the formulas.
  • ✗Using the wrong formula for S_n or a_n, or mixing parts of different formulas.
  • ✗Forgetting to find 'n' first when the last term (a_n) is given and the sum (S_n) is required.

Exam tips

  • ★Carefully read the question to identify the given values (a, d, n, a_n, S_n) and what needs to be found.
  • ★Always write down the correct formula before substituting values to avoid errors and gain partial marks.
  • ★Show all steps clearly, especially for algebraic manipulation, as this helps in identifying and correcting mistakes.
  • ★Double-check all calculations, particularly with signs and multiplication/division.
  • ★Practice a variety of problems, including word problems, to understand the application of A.P. concepts in different contexts.

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