Question

What is the next term to this series, $2,3,7,16,32, \,and \,57....$ ?

Answer 1

Hint: For finding the next term of the series we have observed the pattern , here we have a pattern that is on adding consecutive square numbers to the preceding term , we get the next term very easily .

Complete step by step solution:
we rely on adding consecutive square numbers, the formula for the $nth$ term can be obtained using the method of differences between terms ,
\[2{\text{ }}3{\text{ }}7{\text{ }}16{\text{ }}32{\text{ }}57\] gives
$
  2 + {(1)^2} = 3 \\
  3 + {(2)^2} = 7 \\
  7 + {(3)^2} = 16 \\
  16 + {(4)^2} = 32 \\
  32 + {(5)^2} = 57 \\
 $
\[1{\text{ }}4{\text{ }}9{\text{ }}16{\text{ }}25\] - the pattern of square numbers
This result in \[3{\text{ }}5{\text{ }}7{\text{ }}9\] for the differences between these square numbers, And finally, \[2{\text{ }}2{\text{ }}2\] .
When the third set of differences is constant, the highest term is ${n^3}$ and therefore the number of ${n^3}$ is that this number divided by $6$ .
So the formula starts $\dfrac{{2{n^3}}}{6}$ or $\dfrac{{{n^3}}}{3}$ ,
If you then subtract $\dfrac{{{n^3}}}{3}$ from the first sequence you'll be able to repeat this process to seek out the ${n^2}$ component so on..
The full formula for the $nth$ term is
$\dfrac{{\left( {2{n^3} - 3{n^2} + n + 12} \right)}}{6}$
So the $7th$ term is
$ \Rightarrow \dfrac{{\left( {2{n^3} - 3{n^2} + n + 12} \right)}}{6}$
$
   = \dfrac{{\left( {2{{(7)}^3} - 3{{(7)}^2} + 7 + 12} \right)}}{6} \\
   = \dfrac{{2 \times 343 - 3 \times 49 + 7 + 12}}{6} \\
   = \dfrac{{686 - 147 + 7 + 12}}{6} \\
   = \dfrac{{558}}{6} \\
   = 93 \\
 $
This may look like time taking compared to noticing you’re adding a square number anytime, but if you had to seek out the $100th$, the generic formula eventually becomes quicker.
For the record, the $100th$ term would be ,
$ \Rightarrow \dfrac{{\left( {2{n^3} - 3{n^2} + n + 12} \right)}}{6}$
$
   = \dfrac{{\left( {2{{(100)}^3} - 3{{(100)}^2} + 100 + 12} \right)}}{6} \\
   = \dfrac{{2 \times 1000000 - 3 \times 10000 + 100 + 12}}{6} \\
   = \dfrac{{2000000 - 30000 + 100 + 12}}{6} \\
   = \dfrac{{1997112}}{6} \\
   = 332852 \\
 $
Note: Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily. For solving this type of question, you must understand the pattern and then try to form a general formula so that we can get any term without depending on previous output.