Question

State whether the following sequence is an A.P. or not.1, 3, 6, 10, ……..

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Hint: Here we will find the difference between each of the two consecutive terms to check whether the difference is constant or not.
If the difference turns out to be constant then it given sequence is n A.P. otherwise it is not in A.P.

A.P or arithmetic progression is defined as the sequence of numbers in which the difference between each of the two consecutive terms is constant.

The given sequence is:-

1, 3, 6, 10, ……..

Let the first term be $a_1$
Hence, $a_1 = 1$
Let the second term be $a_2$
Hence,$a_2 = 3$
Let the third term be $a_3$
Hence, $a_3 = 6$

Now let us find the difference between first and the second term:-
$d = a_2 - a_1$
Putting in the respective values we get:-
$\begin{gathered} d = 3 - 1 \\ \Rightarrow d = 2.................\left( 1 \right) \\ \end{gathered}$

Now let us find the difference between the second term and the third term:-
$d = a_3 - a_2$
Putting in the respective values we get:-
$\begin{gathered} d = 6 - 3 \\ \Rightarrow d = 3..................\left( 2 \right) \\ \end{gathered}$
Now since the value of d in equation 1 and equation 2 is not same
Hence we can conclude that the difference is not constant
Therefore,
The given sequence is not in A.P

Note: Students should take note that in an A.P. the common difference of each of the two consecutive terms is equal.
Nth term can be calculated by the formula:-
$T_n = a + \left( {n - 1} \right)d$
Where,
a is the first term of the A.P
n is the number of term in A.P
d is the common difference.