Question 1125867
1.
{{{-9}}},{{{ -8}}},{{{ -4}}},{{{ 5}}},{{{ 21}}},{{{ 46}}} ... ...

Find nth term formula using differences.

Differences:

{{{-9 }}}| |{{{ -8}}} | |{{{ -4}}} | |{{{ 5 }}}| | {{{21}}} | | {{{46}}}
..... | {{{1}}} | |{{{ 4}}} | | {{{9}}} | | {{{16}}} | |{{{ 25}}} | 
............ | | {{{3 }}}| |{{{ 5}}} | |{{{ 7}}} | | {{{9}}} | | 
............. | | |{{{ 2 | |{{{ 2 }}}| | {{{2 }}}| | | 


since we have three differences, the general form for finding the nth term in a cubic sequence is 

{{{a[n]=an^3 + bn^2 + cn + d}}}

Set up a system of four equations with four variables to find the coefficients {{{a}}}, {{{b}}}, {{{c}}} and {{{d}}}. 

Use the values given in the sequence as if they were points on a graph in the form ({{{n}}}, {{{nth}}}). 
It is easiest to start with the first {{{4}}} terms, as they are usually smaller or simpler numbers to work with.

so, we have: 
({{{1}}}, {{{-9}}}), ({{{2}}},{{{ -8}}}), ({{{3}}},{{{-4}}}), ({{{4}}},{{{ 5}}}) 

Plug in to: 

{{{a[n]=an^3 + bn^2 + cn + d}}}

({{{1}}}, {{{-9}}})

{{{-9=a*1^3 + b*1^2 + c*1 + d}}}
{{{-9=a+ b + c + d}}}...............eq.1

({{{2}}},{{{ -8}}})

{{{-8=a*2^3 + b*2^2 + c*2 + d}}}
{{{-8=8a+ 4b + 2c + d}}}...............eq.2


({{{3}}},{{{-4}}})

{{{-4=a*3^3 + b*3^2 + c*3 + d}}}
{{{-4=27a+ 9b + 3c + d}}}...............eq.3

({{{4}}},{{{ 5}}})

{{{5=a*4^3 + b*4^2 + c*4 + d}}}
{{{5=64a+ 16b + 4c + d}}}...............eq.4


Solve the system of {{{4}}} equations using your favorite method.


In this example, the results are: 

{{{a = 1/3}}}, {{{b = -1/2}}}, {{{c = 1/6}}}, {{{d = -9}}}

then we have

{{{a[n]=(1/3)n^3 -(1/2)n^2 + (1/6)n -9}}}

and next term will be:

{{{n=7}}}

{{{a[7]=(1/3)7^3 -(1/2)7^2 + (1/6)7 -9}}}

{{{highlight(a[7]=82)}}}

here are some more terms:

{{{-9}}},{{{ -8}}},{{{ -4}}},{{{ 5}}},{{{ 21}}},{{{ 46}}} {{{82}}},{{{ 131}}},{{{ 195}}},{{{276}}},{{{ 376}}},{{{ 497}}}.............


2.

{{{2}}}, {{{4}}}, {{{6}}}, {{{10}}} ,{{{16}}}, {{{26}}},{{{ 42}}}, ... .

track the sequence of {{{adding}}} the previous {{{2}}} numbers combined:

{{{2+4 =6}}}

{{{4+6=10}}}

{{{6+10=16}}}

{{{10+16=26}}}

{{{16+26=42}}}

so, the next term will be:

{{{26+42=68}}}

your answer is:{{{ 68}}}


some more terms:

{{{2}}}, {{{4}}}, {{{6}}}, {{{10}}}, {{{16}}}, {{{26}}},{{{ 42}}}, {{{68}}}, {{{110}}}, {{{178}}}, {{{288}}}, {{{466}}}, {{{754}}}, {{{1220}}}, ...