Question

Write the next three terms and the rule for the following sequence of numbers:
(a) 3, 4, 6, 8, 12, 14, 18, ___, ___, ___
(b) 2, 6, 12, 20, ___, ___, ___

Answer 1

Hint: Here, we will observe and rewrite the given numbers to find the number pattern/rule of the sequence. Then, we will use the number pattern to write the next three terms of the given sequences.

Complete step-by-step answer:
(a)
First, we will list the first few prime numbers.
The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, etc.
We can observe that the sum of the first prime number 2 and 1 is 3, the sum of the second prime number 3 and 1 is 4, etc.
Adding 1 to each of the listed prime numbers, we get the sequence
3, 4, 6, 8, 12, 14, 18
We can observe that this is the sequence given.
Therefore, the given sequence is formed by adding 1 to prime numbers.
We can find the next three terms by listing the next 3 prime numbers and adding 1 to them.
The next three prime numbers after 17 are 19, 23, 29.
Adding 1 to 19, we get the number 20.
Adding 1 to 23, we get the number 24.
Adding 1 to 29, we get the number 30.
Therefore, the next three terms of the given sequence are 20, 24, and 30.
The sequence becomes 3, 4, 6, 8, 12, 14, 18, 20, 24, 30.
(b)
We will find the difference between two successive numbers in the given sequence.
The difference between the second number and the first number $$=6-2=4$$.
The difference between the third number and the second number $$=12-6=6$$.
The difference between the fourth number and the third number $$=20-12=8$$.
We can observe that the difference between two successive numbers starts from 4, and increases by 2 each time.
Therefore, the given sequence is formed by adding the difference of a number and the previous number, and 2 to the number.
We can find the next three terms by using the rule.
The difference between the fifth number and the fourth number $$=8+2=10$$.
Therefore, we get
Fifth number of the sequence $$=$$ Fourth number of the sequence $$+$$ 10
Thus, we get the fifth number as
Fifth number of the sequence $$=20+10=30$$
The difference between the sixth number and the fifth number $$=10+2=12$$.
Therefore, we get
Sixth number of the sequence $$=$$ Fifth number of the sequence $$+$$ 12
Thus, we get the sixth number as
Sixth number of the sequence $$=30+12=42$$
The difference between the seventh number and the sixth number $$=12+2=14$$.
Therefore, we get
Seventh number of the sequence $$=$$ Sixth number of the sequence $$+$$ 14
Thus, we get the seventh number as
Seventh number of the sequence $$=42+14=56$$
Therefore, the next three terms of the given sequence are 30, 42, 56.
The sequence becomes 2, 6, 12, 20, 30, 42, 56.

Note: We used the term ‘prime numbers’ in the solution. Prime numbers are the numbers which have only two factors, 1 and the number itself. For example: 2 is the product of 2 and 1. Therefore, the factors of 2 are 2 and 1. Thus, 2 is a prime number.
It is possible that the number sequences may have more than one rule. The next three terms using the different rule may be different which is not correct.