Question

How do you write the next $4$ terms in each pattern and write the pattern rule given $8\,,\,12\,,\,24\,,\,60\,,\,168\,$?

Answer 1

Hint: Generally, whenever a pattern is involved in the question we can think of A.P and G.P. In this question first we can check the difference of the given pattern and then we can check the ratio of two successive terms. We can write the next four terms using that rule only.

Complete step-by-step answer:
In the given question,
$8\,,\,12\,,\,24\,,\,60\,,\,168\,$
Notice that all the terms are divisible by $4$, so consider the sequence formed by dividing by $4$ .
$ \Rightarrow \,2\,,\,3\,,\,6\,,\,15\,,\,45$
Write down the sequence of difference between consecutive terms:
$ \Rightarrow \,1\,,\,3\,,\,9\,,\,27$
Notice that these are just powers of $3$ or we can say that they are in G.P.
Hence we can deduce a recursive rule for the original sequence:
$ \Rightarrow \,{a_1}\, = \,8$
$ \Rightarrow \,{a_{n + 1}}\, = \,3{a_n} - 12$
This sequence must also have a general formula of the form:
$ \Rightarrow \,{a_n}\, = \,A.\,{3^{n - 1}}\, + \,B$
Where A, B are constants to be determined.
Where A, B are constants to be determined.
Putting $n\, = \,1\,,\,2$ to get two equations to solve,
We find
$ \Rightarrow \,A + B = 8$
$ \Rightarrow 3A + B = 12$
Subtracting the second equation from $3\, \times $ the first equation, we find
$ \Rightarrow 2B\, = \,12$
Hence $B\, = \,6$ and $A\, = \,2$
So, the general term of our sequence may be written:
${a_n}\, = \,2.\,{3^{n - 1}}\, + 6$
Using recursive formula, we can find the next $4$ terms:
$ \Rightarrow {a_6} = 2.{\left( 3 \right)^5} + 6 = \,286 + 6 = 292$
$ \Rightarrow {a_7} = 2.{\left( 3 \right)^6} + 6 = \,1458 + 6 = 1464$
$ \Rightarrow {a_8} = 2.{\left( 3 \right)^7} + 6 = \,4374 + 6 = 292$
$ \Rightarrow {a_9} = 2.{\left( 3 \right)^8} + 6 = \,13122 + 6 = 13128$

Note: In this question using the first term and the second term we can find the general term of the given sequence to find all the consecutive terms. Generally, the concepts of A.P and G.P are used in these types of questions. Remember the pattern has to be analyzed for all the terms only then it can be concluded for the pattern for the entire series.