Question

How do you find the next number in this sequence \[3,7,14,32,58..........................\] ?

Answer 1

Hint:
Arithmetic sequences: This is a type of sequence where the difference between consecutive terms is always the same and we call that difference a common difference.
$d = {\text{difference of successive terms}}$
Geometric sequence: This is a type of sequence where the ratio between consecutive terms is always the same and we call that ratio a common ratio.
$r = {\text{ratio of successive terms}}$
Now with the help of the above two definitions we can find out whether the given sequence is arithmetic, geometric or neither.

Complete step by step solution:
Given
\[3,7,14,32,58..........................\;\;\;\;\;\;\;\;\;\;...............\left( i \right)\]
Now our aim is to find out whether the given sequence is arithmetic, geometric or neither and proceed accordingly.
So let’s take two cases and assume first it to be arithmetic and then geometric to see which of the conditions it satisfies.
Case I:
Now let’s take the common difference of the successive terms of the given sequence \[3,7,14,32,58..........................\] such that, we can write:
$
  d = 7 - 3 = 4 \\
  d = 14 - 7 = 7 \\
  d = 32 - 14 = 18 \\
  d = 58 - 32 = 26 \\
 $
So on observing the above results we can conclude that they do not have a common difference such that the given sequence is not an arithmetic sequence.
Now let’s take
Case II:
We have the basic definition of geometric sequence as:
Now let’s take the common ratio of the successive terms of the given sequence \[3,7,14,32,58..........................\] such that, we can write:
$
  r = \dfrac{7}{3} \\
  r = \dfrac{{14}}{7} = 2 \\
  r = \dfrac{{32}}{{14}} = \dfrac{{16}}{7} \\
  r = \dfrac{{58}}{{32}} = \dfrac{{29}}{{16}} \\
 $
So on observing the above results we can conclude that they do not have a common ratio such that the given sequence is not a geometric sequence.
So the sequence \[3,7,14,32,58..........................\] is neither arithmetic nor geometric in nature.
But on close observation we can see a pattern such that we can find the next term:
$
  {1^2} + 2 = 3 \\
  \; \swarrow \\
  {2^2} + 3 = 7 \\
  \; \swarrow \\
  {3^2} + 5 = 14 \\
  \; \swarrow \\
  {5^2} + 7 = 32 \\
  \; \swarrow \\
  {7^2} + 9 = 58 \\
  \; \swarrow \\
  {9^2} + 11 = 92 \\
 $

There we can say that the next number in this sequence \[3,7,14,32,58..........................\] would be 92.

Note:
If sequences like the above are given which are neither arithmetic nor geometric in nature, we have to follow the above prescribed method.
Also, General terms of arithmetic and geometric series:
Arithmetic series: ${a_n} = a + \left( {n - 1} \right)d$
Geometric series: ${a_n} = a \times {r^{n - 1}}$