Question

Complete the given series 4,9,13,22,35, ………………

Answer 1

The given series is a special series. So no particular formula exists for this series. The terms of the series are such that every next term is the sum of its previous two terms.

Complete step by step solution: Now, in this question, the first term given to us is 4. The second term is 9 and so on. The third term given is 13. So every next term is the sum of its previous two terms. This is what makes it a special series. Now, the difference between the first and the second term is 5.
\Rightarrow 9 - 4 = 5 \\\ \Rightarrow 9 = 5 + 4 \\\.
The third term is 13 and the difference between the third and second term is 4.
\Rightarrow 13 - 9 = 4 \\\ \Rightarrow 13 = 4 + 9 \\\
The fourth term is 22 and the difference between 13 and 22 is 9.
\Rightarrow 22 - 13 = 9 \\\ \Rightarrow 22 = 9 + 13 \\\
The fifth term is 35 and the difference between 22 and 35 is 13.
\Rightarrow 35 - 22 = 13 \\\ \Rightarrow 35 = 13 + 22 \\\
So to find the sixth term, let the sixth term be ‘x’.
Then the difference between the fifth and sixth term will be 22, such that:
\Rightarrow x - 22 = 35 \\\ \Rightarrow x = 35 + 22 \\\ \Rightarrow x = 57 \\\
Hence, the next number after 35 will be 57. However, the given series is an infinite series. So we can calculate the next terms by applying the same rule. Every next term is the sum of its previous two terms.
So, the term after 57 will be:
$35 + 57 = 92$ and so on.
Therefore, the terms after 35 will be 57, 92 and so on till infinity.

Note: Before starting with the calculations the terms already given in the series have to be observed very carefully first to establish the logic and relation between the terms and then proceed further. There are many such special series with powers also. For example,$1,4n,7{n^2},10{n^3},............$ . They are called Arithmetico-geometric-progression.