AP Formula Class 10

Arithmetic progression( AP) is the sequence of numbers in which the difference between two consecutive terms is equal. AP formulas class 10 are useful in solving problems related to sequence and series concepts. These formulas may seem confusing to students due to the use of various terms such as first term, common difference, nth term and so on. This article covers all the arithmetic progression formulas along with relevant examples and memorization tips to instill a clear understanding of AP formula class 10 and its practical applications.

List of AP Formula Class 10

Here is a list of important AP formulas class 10 and terms related to them that students must learn and understand:

• Arithmetic Sequence in general form: a, a + d, a + 2d, a + 3d, . . .
• AP Formula to find common difference: d = a₂ - a₁
• Arithmetic progression formula for nth term: an = a + (n - 1) d, where ‘a’ depicts the constant term, ‘n’ is the number of terms and ‘d’ is the common difference of the AP.
• The sum of first ‘n’ terms of arithmetic progression when the nth term is NOT known is: S_n = n/2[2a+(n - 1) d]
• The sum of first n terms of an arithmetic progression when the nth term, an is known: is S_n = n/2[a₁+ a_n]

Applications of AP Formula Class 10

Application of AP formula class 10 is useful in the study of different patterns and sequences that exist in the actual world. Here are some practical examples of the applications of arithmetic progression formulas:

• AP formula class 10 allows students to investigate the various forms of patterns and sequences that they must have come across in their daily lives, such as the patterns followed by petals of flowers or spirals on a pine cone.

• These formulas are also applied to determine amounts and values in different scenarios. For example, during a taxi ride, customers are charged an initial amount plus a fixed amount per Km or per mile. Thus, AP formula can be applied to calculate the taxi fare.

• The use of AP formulas is prominent in research work in various sciences such as nuclear science and space science.

AP Formula Class 10 Examples

Example 1: Sunder plans to make a Christmas tree and decides to have many layers of decorations on it. Sunder wants to put 24 objects in the lowest layer and every layer must have three objects more than the layer on its top. If Sunder has 78 objects, how many layers of a Christmas tree can he decorate?

Solution: The arithmetic progression of the decoration layer would be: 24, 21, 18...

As we know that the total number of objects is 78, by using the sum of arithmetic progression formula. S_n = n/2 [2a + (n - 1)d]

We get, S_n= n/2( 2×24 + (n - 1) -3) = 78, where a = 24 and d = - 3

= n/2 ( 48 - 3n + 3) = 78

= 24n - 3/2 n² + 3/2 n = 78

= - 3n² + 51n = 156

= 3n² – 51n + 156 = 0

= n² – 17n + 52 = 0

= (n - 4) ( n - 13) = 0

= n = 4

Therefore, Sunder can decorate 4 layers of Christmas tree using 78 objects in the sequence he wishes.

Please note that for n = 13, the count of objects will be negative numbers which are incorrect and hence it is discarded.

Example 2: In a series, the nth term is 4 + 3n, find the sum of this series of length 10.

Solution: The nth term of AP with first term ‘a’ and common difference ‘d’ is a + (n-1) × d

Therefore, 3n+4 = a-d + nd

By comparing terms we get,

d=3 and a-d = 4

=> a = 7

Therefore, in this series, the difference is 3 and the first element is 7 and the total length of the series is 10.

Using the Arithmetics progression sum formula, n/2 [2a + (n - 1)d], we get S₁₀ = 10/2 (2×7 + (10 -1) 3 )

= 5 ( 14 + 27) = 5 × 41

= 205

Hence, the sum of the first ten elements of the series is 205.

Tips to Memorize AP Formula Class 10

Students can follow the below-mentioned tips to memorize the AP formula class 10:

• The first step is to follow a step-by-step explanation and application of each formula. This will ensure thorough learning of the logic behind each formula thus enabling students to memorize them better.
• Taking random revision tests by themselves will help students to figure out which topics need improvement and attention.
• Students can also use resources like formula sheets or assessment tests available on the internet for the revision of AP formulas.

Students can download the printable Maths Formulas Class 10 sheet from below.