In an AP the sum of
second & third term is 22 & the
Product of first & fourth term is
85. Then find th

1 thought on “In an AP the sum of<br />second & third term is 22 & the<br />Product of first & fourth term is<br />85. Then find th”

1. Answer:

   When the consecutive terms of series differ by a common number, then the series is said to be Arithmetic Progression
   

   
   

   Let a be the first term of the AP
   

   
   

   d be the common difference of the AP
   

   
   

   nth term of AP ⇒ a + ( n – 1) d
   

   
   

   Given,
   

   
   

   The sum of second and third term is 22
   

   
   

   ⇒ (a + d) + (a + 2d) =22
   

   
   

   ⇒ 2a + 3d = 22
   

   
   

   ⇒ d = 1/3 ( 22 – 2a)
   

   
   

   The product of first and fourth term is 85
   

   
   

   ⇒ a ( a + 3d) = 85
   

   
   

   ⇒ a² + 3ad = 85
   

   
   

   Substituting the value of d gives,
   

   
   

   ⇒ a ( a + 3 ( 1/3 * (22 – 2a) )) = 85
   

   
   

   ⇒ a ( a + 22 – 2a) = 85
   

   
   

   ⇒ a ( – a + 22) = 85
   

   
   

   ⇒ – a² + 22a = 85
   

   
   

   ⇒ a² – 22a + 85 = 0
   

   
   

   ⇒ a² – 17a – 5a + 85 = 0
   

   
   

   ⇒ a ( a – 17) – 5 ( a – 17)= 0
   

   
   

   ⇒ (a-5)(a-17)= 0
   

   
   

   ⇒ a = 5 or a = 17
   

   
   

   If a = 5,
   

   
   

   d = 1/3 ( 22 – 10) = 1/3 ( 12) = 4
   

   
   

   If a = 17,
   

   
   

   d = 1/3 ( 22 – 34) = 1/3 ( – 12) = – 4
   

   
   

   a = 5, d = 4
   

   
   

   Then Arithmetic Progression is
   

   
   

   5, 9, 13, 17
   

   
   

   a = 17, d = – 4
   

   
   

   Then Arithmetic Progression is,
   

   
   

   17, 13, 9, 5
   

   
   

   Therefore, The required terms in the AP are 5, 9, 13, 17.

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