Is the following statement true or false?

If the list a, 2a, 3a, 4a ... is an AP then its common difference is a and the next three numbers are 5a, 6a, 7a.

Question

Is the following statement true or false?If the list a, 2a, 3a, 4a ... is an AP then its common difference is a and the next three numbers are 5a, 6a, 7a.

Accepted Answer

Given, the series a, 2a, 3a, 4a..... is an AP.The formula for common difference,

d=
    a
    n
   
   −
    a

n−1

.The common difference between the second and first term is,

⇒
      a
      2
     
     −
      a
      1
     
     =2a−a

⇒
      a
      2
     
     −
      a
      1
     
     =a

The common difference between the third and second term is,

⇒
      a
      3
     
     −
      a
      2
     
     =3a−2a

⇒
      a
      3
     
     −
      a
      2
     
     =a

The common difference between the fourth and third term is,

⇒
      a
      4
     
     −
      a
      3
     
     =4a−3a

⇒
      a
      4
     
     −
      a
      3
     
     =a

Thus, the difference between each term is constant so it is an AP.The fifth term will be,

a
      5
     
     =
      a
      4
     
     +d

⇒
      a
      5
     
     =4a+a

⇒
      a
      5
     
     =5a

The sixth term will be,

a
      6
     
     =
      a
      5
     
     +d

⇒
      a
      6
     
     =5a+a

⇒
      a
      6
     
     =6a

The seventh term,

a
      7
     
     =
      a
      6
     
     +d

⇒
      a
      7
     
     =6a+a

⇒
      a
      7
     
     =7a

Thus the common difference value is

a
   and the next three terms will be 5a, 6a, 7a which is true.Therefore the correct option is 1.

Is the following statement true or false?

If the list a, 2a, 3a, 4a ... is an AP then its common difference is a and the next three numbers are 5a, 6a, 7a.

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