Question

What is the next number in the series 2, 12, 36, 80, 150,?
A. 263
B. 251
C. 284
D. 252

Answer 1

First we will first find the difference between the terms of the given series and we will find the differences of the obtained new series unless the difference is common. Then we will add back the obtained value to find the required value.

Complete step by step answer:
We are given that the series is 2, 12, 36, 80, 150.

Step 1:
We will find the difference between the first and second terms of the given series, we get

$$\Rightarrow 12 - 2 \\\ \Rightarrow 10 \\\$$

Finding the difference between the second and third terms of the given series, we get

$$\Rightarrow 36 - 12 \\\ \Rightarrow 24 \\\$$

We will now find the difference between the third and fourth terms of the given series, we get

$$\Rightarrow 80 - 36 \\\ \Rightarrow 44 \\\$$

Computing the difference between the fourth and fifth term of the given series, we get

$$\Rightarrow 150 - 80 \\\ \Rightarrow 70 \\\$$

Thus, we have a series of differences of 10, 24, 44, 70.
Let us assume that the fifth term is $x$.

Step 2:
We will find the difference between the first and second terms of the new series, we get

$$\Rightarrow 24 - 10 \\\ \Rightarrow 14 \\\$$

Finding the difference between the second and third terms of the new series, we get

$$\Rightarrow 44 - 24 \\\ \Rightarrow 20 \\\$$

We will now find the difference between the third and fourth terms of the new series, we get

$$\Rightarrow 70 - 44 \\\ \Rightarrow 26 \\\$$

Thus, we have a new series of differences: 14, 20, 26.
Let us assume that the fourth term of the above series is $y$.

Step 3:
We will find the difference between the first and second terms of the new series, we get

$$\Rightarrow 20 - 14 \\\ \Rightarrow 6 \\\$$

Finding the difference between the second and third terms of the new series, we get

$$\Rightarrow 26 - 20 \\\ \Rightarrow 6 \\\$$

Since the difference is 6 for all, so the difference between the third and fourth terms of the new series is 6.
Adding the number 6 in the third term of the series of step 2 to find the value of $y$, we get

$$\Rightarrow y = 26 + 6 \\\ \Rightarrow y = 32 \\\$$

Adding the above number with the fourth term of the series of step 1 to find the value of $x$, we get

$$\Rightarrow x = 70 + 32 \\\ \Rightarrow x = 102 \\\$$

Finding the sum of the above number with the fifth term of the given series to find the required value, we get

$$\Rightarrow 150 + 102 \\\ \Rightarrow 252 \\\$$

Hence, the correct option is D.

Note: In solving these types of questions, you need to know the basic properties and meaning of the geometric progression and arithmetic progression. One should know that the common difference is when the difference between each number in the series is the same. Avoid calculation mistakes.