Question

Given sequence 2,5,8

Prove that none of the terms of this sequence are perfect square

Answer 1

Given sequence is - 2,5,8,....

a=2 ; d=3

An=a+3n

Let,

a2=(a+3n)2

a2=a2+6n+9n2

9n2+6n=0

3(3n2+2)=0

3n2+2=0

3n2=-2

n2=-2/3

n=Root of -2/3

-2/3>0

So, it does not contain any perfect number in this sequence.

Answer 2

No term of the sequence 2 , 5 , 8 is a perfect square

Given :

The sequence 2 , 5 , 8

To find :

To prove no term of the sequence 2 , 5 , 8 is a perfect square

Solution :

Step 1 of 3 :

Write down the given sequence

The given sequence is 2 , 5 , 8

Step 2 of 3 :

Find general term of the sequence

The given sequence is an arithmetic sequence

First term = a = 2

Common Difference = d = 5 - 2 = 3

The nth term of the sequence

= a + ( n - 1 )d

= 2 + ( n - 1 ) × 3

= 2 + 3n - 3

= 3n - 1

Step 3 of 3 :

Prove that no term of the sequence is a perfect square

If possible let nth term of the sequence is a perfect square

Then 3n - 1 = k² for some natural number k

⇒ 3n = k² + 1

Now 3n is divisible by 3

⇒ k² + 1 is divisible by 3

Which is a contradiction

Hence no term of the sequence 2 , 5 , 8 is a perfect square

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