Subscribe to our YouTube channel for the latest videos, updates, and tips.


Problems on Sum of 'n' Terms of Arithmetic Progression

Here we will learn how to solve different types of problems on sum of n terms of Arithmetic Progression.

1. Find the sum of the first 35 terms of an Arithmetic Progression whose third term is 7 and seventh term is two more than thrice of its third term.

Solution:

Let us assume that ‘a’ be the first term and ‘d’ be the common difference of the given Arithmetic Progression.

According to the problem,

3rd term of an Arithmetic Progression is 7

i.e., 3th term = 7

⇒ a + (3 - 1)d = 7

⇒ a + 2d = 7 ................... (i)

and seventh term is two more than thrice of its third term.

i.e., 7th term = 3 × 3rd term + 2

⇒ a + (7 - 1)d = 3 × [a + (3 - 1)d] + 2

⇒ a + 6d = 3 × [a + 2d] + 2

Substitute the value of a + 2d = 7 we get,

⇒ a + 6d = 3 × 7 + 2

⇒ a + 6d = 21 + 2

⇒ a + 6d = 23 ................... (ii)

Now, subtract the equation (i) from (ii) we get,

4d = 16

⇒ d = \(\frac{16}{4}\)

⇒ d = 4

Substitute the value of d = 4 in the equation (i) we get,

⇒ a + 2 × 4 = 7

⇒ a + 8 = 7

⇒ a = 7 - 8

⇒ a = -1

Therefore, the first term of the Arithmetic Progression is -1 and common difference of the Arithmetic Progression is 4.

Now, sum of the first 35 terms of an Arithmetic Progression S\(_{35}\) = \(\frac{35}{2}\)[2 × (-1) + (35 - 1) × 4], [Using the Sum of the First n Terms of an Arithmetic Progression S\(_{n}\) = \(\frac{n}{2}\)[2a + (n - 1)d]

\(\frac{35}{2}\)[-2 + 34 × 4]

\(\frac{35}{2}\)[-2 + 136]

\(\frac{35}{2}\)[134]

= 35 × 67

= 2345.

 

2. If the 5th term and 12th term of an Arithmetic Progression are 30 and 65 respectively, find the sum of its 26 terms.

Solution:

 Let us assume that ‘a’ be the first term and ‘d’ be the common difference of the given Arithmetic Progression.

According to the problem,

5th term of an Arithmetic Progression is 30

i.e., 5th term = 30

⇒ a + (5 - 1)d = 30

⇒ a + 4d = 30 ................... (i)

and 12th term of an Arithmetic Progression is 65

i.e., 12th term = 65

⇒ a + (12 - 1)d = 65

⇒ a + 11d = 65 .................... (ii)

Now, subtract the equation (i) from (ii) we get,

7d = 35

⇒ d = \(\frac{35}{7}\)

⇒ d = 5

Substitute the value of d = 5 in the equation (i) we get,

a + 4 × 5 = 30

⇒ a + 20 = 30

⇒ a = 30 - 20

⇒ a = 10

Therefore, the first term of the Arithmetic Progression is 10 and common difference of the Arithmetic Progression is 5.

Now, sum of the first 26 terms of an Arithmetic Progression S\(_{26}\) = \(\frac{26}{2}\)[2 × 10 + (26 - 1) × 5], [Using the Sum of the First n Terms of an Arithmetic Progression S\(_{n}\) \(\frac{n}{2}\)[2a + (n - 1)d]

= 13[20 + 25 × 5]

= 13[20 + 125]

= 13[145]

= 1885

Arithmetic Progression




11 and 12 Grade Math 

From Problems on Sum of 'n' Terms of Arithmetic Progression to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Rounding Decimals | Questions Related to Round a Decimal

    May 14, 25 04:21 PM

    The worksheet on rounding decimals would be really good for the students to practice huge number of questions related to round a decimal. This worksheet include questions related

    Read More

  2. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 14, 25 03:01 PM

    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More

  3. Worksheet on Rounding Off Number | Rounding off Number | Nearest 10

    May 14, 25 12:50 PM

    In worksheet on rounding off number we will solve 10 different types of problems. 1. Round off to nearest 10 each of the following numbers: (a) 14 (b) 57 (c) 61 (d) 819 (e) 7729 2. Round off to

    Read More

  4. Rounding Off to the Nearest Whole Number | Nearest 10, 100, and 1000

    May 13, 25 03:43 PM

    Nearest Ten
    Here we will learn how to rounding off to the nearest whole number?

    Read More

  5. Conversion of Improper Fractions into Mixed Fractions |Solved Examples

    May 12, 25 04:52 AM

    Conversion of Improper Fractions into Mixed Fractions
    In conversion of improper fractions into mixed fractions, we follow the following steps: Step I: Obtain the improper fraction. Step II: Divide the numerator by the denominator and obtain the quotient…

    Read More