Ex 5.3 Q5 - The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Ex 5.3 Q5 - The first term of an AP is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

Solution:-

Let,

      First term = a

      Common difference = d

      Number of terms = n

      Last term = aₙ

Given,

      First term = a = 5

      Last term = aₙ = 45

      Sum of n terms = Sₙ = 400

➙ We know that, the sum of n terms of an AP is given by,

      Sₙ =

n / 2

(a + aₙ)

   ∴ 400 =

n / 2

(5 + 45)

   ∴ 400 =

n / 2

(50)

   ∴ 400 = n(25)

   ∴ n =

400 / 25

   ∴ n = 16 

Therefore, there are 16 terms in the given AP.

➙ We know that, the nth term of an AP is given by,

      aₙ = a + (n - 1)d

   ∴ 45 = 5 + (16 - 1)d

   ∴ 45 = 5 + 15d

   ∴ 15d = 45 - 5

   ∴ 15d = 40

   ∴ d =

40 / 15

   ∴ d =

8 / 3

Therefore, the common difference of the given AP is 8/3.

Hence,

          The number of terms and the common difference is 16 and 8/3 respectively.

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Try This..

✒ The 7th term of an AP is 35 and the 12th term is 60. Find the first term and the common difference.

✒ The sum of all natural numbers between 10 and 50 which are divisible by 5.

✒ If the sum of the first 20 terms of an AP is 210 and the first term is 5, find the common difference.

✒ The first term of an AP is 10, the last term is 50, and the sum is 480. Find the number of terms and common difference.

❌ Common Mistakes

Mixing up formulas:

▸ Use Sₙ = n/2(a + aₙ) when last term is given; not the one with d!

Forgetting order of operations:

▸ Always simplify brackets and fractions correctly.

Rushing to solve:

▸ Always double-check your substitution and simplification.

▸ Slow down, write neatly, and think through each step; math rewards clarity!

📝 Related Questions:-

• Ex 5.3 Q4 - How many terms of the AP: 9, 17, 25,... must be taken to give a sum of 636?
• Ex 5.3 Q6 - The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
• Ex 5.3 Q15 - A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹200 for the first day, ₹250 for the second day, ₹300 for the third day, etc., the penalty for each succeeding day being ₹50 more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by 30 days?
• Ex 5.3 Q16 - A sum of ₹700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each of the prizes.
• Ex 5.3 Q17 - In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class I will plant 1 tree, a section of Class II will plant 2 trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

Queries Solved:-

Class 10 Ex 5.3

Ex 5.3 Q5 Class 10

Class 10 Ex 5.3 Q5

Class 10 Chap 5 Ex 5.3 Q5

Class 10 Arithmetic Progression

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