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

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Answer 1 · 2016-11-24T07:40:43

The next $4$ terms are:

$492, 1464, 4380, 13128$

Recursive rule is:

${ (a_1 = 8), (a_(n+1) = 3a_n-12) :}$

General formula is:

$a_n = 2*3^(n-1)+6$

Given:

$8, 12, 24, 60, 168$

Notice that all of the terms are divisible by $4$ , so consider the sequence formed by dividing by $4$ :

$2, 3, 6, 15, 42$

Write down the sequence of differences between consecutive terms:

$1, 3, 9, 27$

Notice that these are just powers of $3$ .

Hence we can deduce a recursive rule for the original sequence:

${ (a_1 = 8), (a_(n+1) = 3a_n - 12) :}$

This sequence must also have a general formula of the form:

$a_n = A*3^(n-1) + B$

where $A, B$ are constants to be determined.

Putting $n = 1, 2$ to get two equations to solve, we find:

${ (A+B = 8), (3A+B = 12) :}$

Subtracting the second equation from $3 xx$ the first equation, we find:

$2B = 12$

Hence $B=6$ and $A=2$

So the general term of our sequence may be written:

$a_n = 2*3^(n-1) + 6$

Using the recusive formulation, we can find the next $4$ terms:

$a_6 = 3a_5-12 = 3*168-12 = 492$

$a_7 = 3a_6-12 = 3*492-12 = 1464$

$a_8 = 3a_7-12 = 3*1464-12 = 4380$

$a_9 = 3a_9-12 = 3*4380-12 = 13128$