Given an $A.P.$ whose terms are all positive integers. The sum of its first nine terms is greater than $200$ and less than $220$. If the second term in it is $12$, then its $4^{th}$ term is

If the sum of first $n$ terms of an $A.P.$ is $c n^2$, then the sum of squares of these $n$ terms is

The sequence $\frac{5}{{\sqrt 7 }}$, $\frac{6}{{\sqrt 7 }}$, $\sqrt 7 $, ....... is

The Fibonacci sequence is defined by

$1 = {a_1} = {a_2}{\rm{ }}$ and ${a_n} = {a_{n - 1}} + {a_{n - 2}},n\, > \,2$

Find $\frac{a_{n+1}}{a_{n}},$ for $n=1,2,3,4,5$

The income of a person is $Rs. \,3,00,000,$ in the first year and he receives an increase of $Rs.\,10,000$ to his income per year for the next $19$ years. Find the total amount, he received in $20$ years.

Let the sequence $a_{n}$ be defined as follows:

${a_1} = 1,{a_n} = {a_{n - 1}} + 2$ for $n\, \ge \,2$

Find first five terms and write corresponding series.