Question

Complete the table for the four sequences A, B, C and D.

	Sequence	Next term	nth term
A	2, 5, 8, 11
B	20, 14, 8, 2
C	1, 4, 9, 16
D	0, 2, 6, 12

Answer 1

Hint: In this particular type of question use the concept that first finds out that the given series follows which trend i.e. whether it follows A.P, G.P or any other trend then write its general term according to the trend so use these concepts to reach the solution of the question.

Complete step-by-step answer:
$\left( A \right)$ 2, 5, 8, 11
As we see that the above series follow the property of A.P, with first term (a) = 2, common difference (d) = (5 -2) = (8 – 5) = (11 – 8) = 3
So the ${n^{th}}$ term of the A.P is given by the formula,
$ \Rightarrow {T_n} = a + \left( {n - 1} \right)d$, where ${T_n}$ is the ${n^{th}}$ term and n = 1, 2, 3, 4........
Now substitute the values we have,
$ \Rightarrow {T_n} = 2 + \left( {n - 1} \right)3$
Now simplify this we have,
$ \Rightarrow {T_n} = 2 + 3n - 3 = 3n - 1$
So the $5^{th}$ term or the next term of the series is when n = 5,
$ \Rightarrow {T_5} = 3\left( 5 \right) - 1 = 15 - 1 = 14$
$\left( B \right)$ 20, 14, 8, 2
As we see that the above series follow the property of A.P, with first term (a) = 20, common difference (d) = (14 -20) = (8 – 14) = (2 – 8) = -6
So the ${n^{th}}$ term of the A.P is given by the formula,
$ \Rightarrow {T_n} = a + \left( {n - 1} \right)d$, where ${T_n}$ is the ${n^{th}}$ term and n = 1, 2, 3, 4........
Now substitute the values we have,
$ \Rightarrow {T_n} = 20 + \left( {n - 1} \right)\left( { - 6} \right)$
Now simplify this we have,
$ \Rightarrow {T_n} = 20 - 6n + 6 = 26 - 6n$
So the $5^{th}$ term or the next term of the series is when n = 5,
$ \Rightarrow {T_5} = 26 - 6\left( 5 \right) = 26 - 30 = - 4$
$\left( C \right)$ 1, 4, 9, 16
So as we observe the series carefully we can say that the above series follow the trend of square of first natural numbers.
i.e. ${1^2},{2^2},{3^2},{4^2}....$
So the ${n^{th}}$ term of the series is ${n^2}$, where, n = 1, 2, 3, 4....
\[ \Rightarrow {T_n} = {n^2}\]
So the $5^{th}$ term or the next term of the series is when n = 5,
\[ \Rightarrow {T_5} = {5^2} = 25\]
$\left( D \right)$ 0, 2, 6, 12
So as we observe the series carefully we can say that the above series follow the trend of n (n – 1), where, n = 1, 2, 3, 4....
So the ${n^{th}}$ term of the series is n (n – 1)
\[ \Rightarrow {T_n} = n\left( {n - 1} \right)\]
So the $5^{th}$ term or the next term of the series is when n = 5,
\[ \Rightarrow {T_5} = 5\left( {5 - 1} \right) = 5\left( 4 \right) = 20\]
So the above table become,

	Sequence	Next term	nth term
A	2, 5, 8, 11	14	3n – 1
B	20, 14, 8, 2	-4	26 – 6n
C	1, 4, 9, 16	25	${n^2}$
D	0, 2, 6, 12	20	n(n – 1)

So this is the required answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the ${n^{th}}$term formula for the A.P which is stated above, then just substitute the values in this formula as above substituted and simplify we will get the required ${n^{th}}$ term of the series, then which term we wish to calculate just substitute that number in place of n as above we will get the required term.