Question

clearly ,
$a_{n + 1} - a_{n}$
is not independent of 'n' and is therefore not constant . So , the given sequence is not an A.P.

the full question is given above .

Can someone explain the statement i have written .​

Answer 1

Answer:

Now don't do that way just find 4 terms with the help of given formula.

find common difference if they are same then it's an Ap.

or

Just find arithmetic mean , b = (a+c) /2

And for your statement it means that whenever you have common difference then it doesn't have to depend on 'n'. Because common difference is always constant.

a_n = 2 n^2 + 1

then,

a_(n+1) = 2 (n+1) ^2 + 1

And we know that, if Any sequence is in AP then it should be have common difference.

d = [a_(n+1) ] - [ a_n ]

= [ 2 (n+1) ^2 + 1 ] - [ 2 n^2 + 1 ]

= [ 2 n^2 + 2 + 2n + 1 ] - [ 2 n^2 + 1 ]

= 2 n^2 + 2n + 3 - 2 n^2 - 1

= 2n + 2

= 2(n+1)

And we know common difference doesn't depend on 'n'. Because common difference is always to be CONSTANT.

Answer 2

Answer:

This statement tells that in the expression :

$a_{n + 1} - a_{n}$

when the value of n will change the value of the expression will also change.

For eg., when n is 1,2,3 and so on the value of the expression will keep on changing,i.e., the value of the expression is not constant or fixed.

Hope that it helps you.