Question

What is the next term in the sequence: 3, 6, 12, 21, 33…?

Answer 1

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 of the terms. And this pattern is known as sequence.

Complete step-by-step solution:
According to our question we have to find the next term of the sequence 3, 6, 12, 21, 33…. As we know that the sequence of any queue is in the pattern of a fixed rule. And every next digit of that sequence will be written by the rule. If we see in our question then the sequence is given as, 3, 6, 12, 21, 33,… If we want to find the rule, then we subtract one digit from its next digit, So, we get,
$\begin{aligned} & 6-3=3 \\\ & 12-6=6 \\\ & 21-12=9 \\\ & 33-21=12 \\\ \end{aligned}$
So, we find the difference as: 3, 6, 9, 12 and this difference is in the pattern of $3\left( n-1 \right)+{{\left( n+1 \right)}^{th}}$ digit. So, if we check this rule, then,
For $n=1\text{ }{{T}_{n}}=3\left( n-1 \right)+{{\left( n+1 \right)}^{th}}$ digit $\because {{T}_{1}}=3$
For $n=2\text{ }{{T}_{2}}=3\left( 2-1 \right)+3\Rightarrow 3+3=6$
For $n=3\text{ }{{T}_{3}}=3\left( 3-1 \right)+6\Rightarrow 6+6=12$
For $n=4\text{ }{{T}_{4}}=3\left( 4-1 \right)+12\Rightarrow 9+12=21$
For $n=5\text{ }{{T}_{5}}=3\left( 5-1 \right)+21\Rightarrow 12+21=33$
As we can see that all the terms are the same as our question, so the next term would be,
$n=6\text{ }{{T}_{6}}=3\left( 6-1 \right)+33\Rightarrow 15+33=48$
So, the next term of the sequence is 48.

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.