Question

How do you express the sequence below as a recursively defined function 4, 11, 25, 53, 109, ....?

Answer 1

Hint: First mark the points in the given order and find the differences between two consecutive points. From the result of these differences, find a relation between them. Then form the recursively defined function by observing the sequence of points.

Complete step by step answer:
A recursive defined function defines a value of a function at some natural number ‘n’ in terms of the function's value at some previous point(s).
Let the points are:
${{a}_{1}}=4$ , ${{a}_{2}}=11$ , ${{a}_{3}}=23$ , ${{a}_{4}}=53$ …..${{a}_{n}}$
And the differences between two consecutive points are:
${{d}_{1}}={{a}_{2}}-{{a}_{1}}=11-4=7$
${{d}_{2}}={{a}_{3}}-{{a}_{2}}=25-11=14$
${{d}_{3}}={{a}_{4}}-{{a}_{3}}=53-25=28$
From the above result it is clear that the differences are the consecutive multiple of 7.
Thus it can be written that:
${{d}_{1}}=7\times 1$ , ${{d}_{2}}=7\times 2$ , ${{d}_{3}}=7\times 3$ , ${{d}_{n}}=7\times n$ …..
Framing in sequence we get:
\[\begin{align}
  & {{a}_{2}}={{a}_{1}}+{{d}_{1}} \\
 & \Rightarrow {{a}_{2}}={{a}_{1}}+\left( 7\times 1 \right) \\
\end{align}\]
Similarly;
\[\begin{align}
  & {{a}_{3}}={{a}_{2}}+{{d}_{2}} \\
 & \Rightarrow {{a}_{3}}={{a}_{2}}+\left( 7\times 2 \right) \\
\end{align}\]
Observing the above equations we can form the recursively defined function as;
\[\begin{align}
  & {{a}_{n}}={{a}_{n-1}}+{{d}_{n-1}} \\
 & \Rightarrow {{a}_{n}}={{a}_{n-1}}+7\times \left( n-1 \right) \\
\end{align}\]
This function satisfies the given sequence.
So, it is the required solution to the given question.

Note: The expression should be analyzed by taking each term and it’s difference with the consecutive term. By observing that we should develop the recursively defined function as \[{{a}_{n}}={{a}_{n-1}}+{{d}_{n-1}}\] where ‘a’ denotes the terms and ‘d’ denotes difference between two consecutive terms.