Question

What are the next two terms in the sequence: 12,21,30,39,183, 327,.....?

Answer 1

A geometric sequence is one in which the ratio of each term to its preceding term is always constant. This ratio is called common ratio.
In the series,
{
3
,
12
,
21
,
30
,
...
...
...
...
.
}

we have
12
3
=
4
,
21
12
=
7
4
,
30
21
=
10
9

Hence it is not a geometric sequence.
However, we have
12
−
3
=
21
−
12
=
30
−
21
=
9
and hence we have a common difference and hence it is an arithmetic sequence

Answer 2

Answer:

What is the next number of the sequence 3, 12, 21, _, _, _?

The pattern from what is given looks to be that

[math]3\text{,}\space 12\text{,}\space 21\text{,}\space\text{_}\text{,}\space\text{_}\text{,}\space\text{_}\space\text{is}\space [/math]

[math]3\text{,}\space 3+9\text{,}\space 12+9\text{,}\space\text{_}\text{,}\space\text{_}\text{,}\space\text{_}[/math]

[math]\implies 3\text{,}\space 3+9\text{,}\space 3+18\text{,}\space\text{_}\text{,}\space\text{_}\text{,}\space\text{_}[/math]

[math]\implies 3+9×0\text{,}\space 3+9×1\text{,}\space [/math]

[math]3+9×2\text{,}\space\text{_}\text{,}\space\text{_}\text{,}\space\text{_}\text{.}[/math]

So, continued on indefinitely, the [math]n^{\underline{th}}[/math] number of the given sequence is [math]3+9n\text{.}[/math]

So the 4th number is obtained by letting [math]n=4[/math] in the formula:

[math]3+4×9=3+36=39\text{.}[/math]

Filling in the next two blanks shown in the question, get

[math]3+5×9=3+45=48\space\text{and}\space 3+6×9=3+54=57\text{.}[/math]

Step-by-step explanation: