Question

The sum of first 8 terms of an AP is 100 and sum of first 19 terms is 551. Find the AP ​

Answer 1

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Answer 2

Given :-

• The sum of first 8 terms of an AP is 100 and sum of first 19 terms is 551. Find the AP ​

Solution :-

• Before stepping into the solution, let us know what is an AP. An AP( Arithmetic progression) is a sequence of numbers where the difference between two successive numbers is always the same. The first term is denoted by a and the common difference between every two successive numbers is denoted by d. So, if we know a & d, we can obtain the AP by going on adding d to every number, starting from the first term i.e a . For example, if the first term(a) = 1 and common difference is 2, then the AP is 1 , 1+2 , 1+2+2  = 1,3,5 and so on..

• First of all, let us consider the first case i.e sum of first 8 terms of an AP is 100

      Here,

      → n = 8

      → Sₙ = 100

• We know that, sum of first n terms of an AP $\sf\ = \frac{n}{2}(2a + (n-1)d)$

       ⇒ 100 = $\sf\frac{8}{2}$(2a + (8-1)d)

       ⇒ 100 = 4(2a + 7d)

       ⇒ 4(25) = 4(2a + 7d)

       ⇒ 2a + 7d = 25     --------------------- (1)

• Now, consider the second case i.e sum of first 19 terms is 551.

       Here,

      → n = 19

      → Sₙ = 551

• Substitute the values in the same formula

       ⇒ 551 = $\sf\frac{19}{2}$(2a + (19-1)d)

       ⇒ 551 = $\sf\frac{19}{2}$(2a + 18d)

       ⇒ 551 = $\sf\frac{19}{2}$(2)(a + 9d)

       ⇒ 551 = 19(a + 9d)

       ⇒ 19(29) = 19(a + 9d)

       ⇒ 29 = a + 9d     ------------------- (2)

• Solving equation (1) & equation (2), we get

     → a = 2

     → d = 3

• So, the AP is  2 , 2 + 3 , 2 + 3 + 3

      = 2 , 5 , 8 , 11 , 14 , 17 , 20 , 23

∴ The AP is 2 , 5 , 8 , 11 , 14 , 17 , 20 , 23 , 26  and so on..

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