Finding the Missing term

What is the next term of the following sequences ?

• 3,8,13,24,41,… Options: {70,75,80,85}.
• 4,23,60,121,… Options: {212,221,241,242}.

This two question is taken from a test paper of mine, I am no able to figure out the correct answer.

For the second one, take a look at the Online Encyclopedia of Integer Sequences[1]. Sequence #A030653 is what you are wanting.

(You can look at [1] for more extensive information including sources, but I will tell a bit about it here.)

Let a(n) be the n-th number of the sequence for n∈{1,2,3…}

Then we have that a(n)=n3+3n2+3n−3

Examples:

a(1)=1+3(1)+3(1)−3=4

a(2)=8+3(4)+3(2)−3=8+12+6−3=23  ……

For the next number in the sequence, we need to calculate a(5). Using the formula above, we get,

a(5)=125+3(25)+3(5)−3=125+75+15−3=212.

So 212 is the next number in the sequence.

Identifying and extending number sequences:

To extend the terms in a sequence, first identify the rule followed by the numbers in the sequence and then using the rule find the next numbers in the sequence.

Example: Find the next three terms in the sequence,

               21, 30, 39, 48, 57, 65, ___, ___, ___

In the above example, the terms in the sequence follow the rule “Add 9”.

     

So the first number after 65 will be 65 + 9 = 74.

The number after 74 will be 74 + 9 = 83.

The number after 83 will be 83 + 9 = 92.

Hence, the next three terms of the sequence will be 74, 83 and 92.

Finding the missing term in a number sequence:

To find the missing term in a number sequence, identify the rule followed by the numbers in the sequence and then use this rule to find the missing term.

Example: Find the missing term in the following sequence

              14, 22, 20, 28, 26, 34, ____, 40, 38

In the above example, the rule followed by the numbers is “Add 8 and then subtract 2”.

    

Here you can see that the last number before the missing number was formed by adding 8. So to find the next term subtract 2 from 34: 34 – 2 = 32

Hence, the missing term in the given number sequence is 32. You can then verify that the next number should be 32 + 8 = 40.