If the sum of 5 terms of an A.P. is same as the sum of its 11 terms, then sum of 16 terms is

The problem requires us to determine the sum of 16 terms in an Arithmetic Progression (AP) given that the sum of its first 5 terms is equal to the sum of its first 11 terms.

Step-by-Step Solution

Step 1: Represent the sums of 5 and 11 terms.

• The sum of the first 5 terms of the AP is given by: S5 = (5/2) [2a + 4d], where a is the first term and d is the common difference.
• The sum of the first 11 terms is given by: S11 = (11/2) [2a + 10d].

Step 2: Equating the two sums:

S₅ = S₁₁    (5/2) [2a + 4d] = (11/2) [2a + 10d]    

Simplify the equation by removing the common factor 1/2:

5 [2a + 4d] = 11 [2a + 10d]    

Expand both sides:

10a + 20d = 22a + 110d    

Rearranging terms gives:

12a = -90d    

Simplify to find a in terms of d:

a = -90d / 12    a = -15d / 2    

Step 3: Find the sum of 16 terms:

• The sum of 16 terms is: S16 = (16/2) [2a + 15d].
• Substitute the value of a: S16 = 8 [2(-15d / 2) + 15d].

• Simplify the terms:

  S₁₆ = 8 [-15d + 15d]            

  S₁₆ = 8 [0]           

  S₁₆ = 0            

Final Answer

The sum of the first 16 terms of the given AP is 0.