What is the rule for the sequence 3, 4, 7, 12?

Hint: For solving this question you should know about the sequence of numbers. We can calculate the sequence rule by subtracting any term from the next term of that, and doing this for 2 – 3 continuous terms and then we will find a fixed pattern of increasing the terms. And this pattern is known as sequence.

Complete step by step solution:
According to our question we have to find the sequence rule for any given sequence: 3, 4, 7, 12.
As we know that the sequence of any queue is in the pattern of fixed rule. And every next digit of that sequence will be written by that rule.
If we see in our question then the sequence is given as: 3, 4, 7, 12, …..
If we subtract one digit from it’s next digit then,
\[\begin{align}
  & 4-3=1 \\
 & 7-4=3 \\
 & 12-7=5 \\
\end{align}\]
So, we found the difference as 1, 3, 5 and this difference is in the pattern of 2n – 1.
So, for the sequence if we check this rule, then for n = 1
\[\begin{align}
  & {{T}_{n}}=2n-1 \\
 & \because n=1,{{T}_{1}}=2\left( 1 \right)-1=2-1=1 \\
\end{align}\]
For \[n=2,{{T}_{1}}=2\left( 2 \right)-1=4-1=3\]
For \[n=3,{{T}_{1}}=2\left( 3 \right)-1=6-1=5\]
So, it is cleared that \[{{T}_{n+1}}-{{T}_{n}}=2n-1\].
And if we add then as \[{{T}_{n+1}}=\left( 2n-1 \right)+{{T}_{n}}\]
Then \[n=1,{{T}_{2}}=1+3=4\]
Since, \[{{T}_{1}}=3\]
\[\begin{align}
  & n=2,{{T}_{3}}=3+4=7 \\
 & n=3,{{T}_{4}}=5+7=12 \\
\end{align}\]
So, it is proved that the rule of sequence is correct and that is \[{{T}_{n+1}}-{{T}_{n}}=2n-1\].

Note: For calculating the sequence of any continuous terms we always find the sequence rule first and then calculate the next values from that. And every sequence has a fixed rule but we have to find it carefully and first apply it for the given terms and then find the other terms.