Question

How do you find the next two terms in 0, 3, 8, 15, 24?

Answer 1

Hint: To solve these types of problems which have a series of numbers, we can find the pattern between the numbers and predict the next term based on this pattern. The pattern can be between the sum or difference or product of consecutive terms.

Complete answer:
We are given the series 0, 3, 8, 15, 24 and we have to find the next two-term in the series. We know that there must be a relation between the terms of the series that we must find to tell the next term.
The first two terms of the series are 0 and 3 respectively. Let’s say the difference between these terms is \[{{d}_{1}}\]. Then \[{{d}_{1}}=3-0=3\].
The second and third terms are 3 and 8 respectively. Let’s say the difference between these terms is \[{{d}_{2}}\]. Then \[{{d}_{2}}=8-3=5\].
The third and fourth terms are 8 and 15 respectively. Let’s say the difference between these terms is \[{{d}_{3}}\]. Then \[{{d}_{3}}=15-8=7\].
Similarly, for the fourth and fifth term difference between them, \[{{d}_{4}}=24-15=9\].
We can see the relationship between the \[{{d}_{1}},{{d}_{2}},{{d}_{3}}\And {{d}_{4}}\] which is, \[{{d}_{2}}={{d}_{1}}+2,{{d}_{3}}={{d}_{2}}+2\And {{d}_{4}}={{d}_{3}}+2\]. We can use this to find the difference between the fifth & sixth and sixth & seventh term. Let’s say the difference between them is \[{{d}_{5}}\And {{d}_{6}}\]. Using the above relation,
\[\begin{align}
  & \Rightarrow {{d}_{5}}={{d}_{4}}+2=9+2=11 \\
 & \Rightarrow {{d}_{6}}={{d}_{5}}+2=11+2=13 \\
\end{align}\]
We can find the sixth and seventh term using these differences as,
Sixth term = Fifth term \[+{{d}_{5}}=24+11=35\] & seventh term = sixth term \[+{{d}_{6}}=35+13=48\].
Hence, the next two terms are \[35\And 48\] respectively.

Note: We can also generate a \[{{n}^{th}}\] term formula from this pattern. This might be useful if we have to find a term with very large \[n\]. For this series the \[{{n}^{th}}\] term is \[{{T}_{n}}={{T}_{n-1}}+2(n+1)+1\]. We can find any term using this general term formula.