What is the next number in the sequence 10, 14, 25, 55, 140?

Steps to Solve

1. Identify the pattern in the sequence

Let's denote the sequence as:

$$ a_1 = 10, \quad a_2 = 14, \quad a_3 = 25, \quad a_4 = 55, \quad a_5 = 140 $$

We will look for a relationship between successive terms.

2. Calculate the differences between the terms

Let's find the differences between consecutive terms:

$$ a_2 - a_1 = 14 - 10 = 4 $$

$$ a_3 - a_2 = 25 - 14 = 11 $$

$$ a_4 - a_3 = 55 - 25 = 30 $$

$$ a_5 - a_4 = 140 - 55 = 85 $$

So we have the first differences:

$$ 4, 11, 30, 85 $$

3. Calculate the second differences

Next, we find the differences of the first differences:

$$ 11 - 4 = 7 $$

$$ 30 - 11 = 19 $$

$$ 85 - 30 = 55 $$

So, the second differences are:

$$ 7, 19, 55 $$

4. Calculate the third differences

Now, we find the differences of the second differences:

$$ 19 - 7 = 12 $$

$$ 55 - 19 = 36 $$

So, the third differences are:

$$ 12, 36 $$

5. Identify a polynomial pattern

The pattern of differences suggests we may be dealing with a polynomial. The third differences are not constant, but they may continue to increase geometrically or arithmetically.

6. Predict the next difference

If we examine the last difference ($36$) and consider a polynomial behavior, we can predict the next difference in a similar increasing manner. Assume the constant increase in third differences leads us to estimate the next term of the sequence.

Continuing the sequence of third differences, we guess:

$$ 12, 36, \text{(next)} = 108 $$

7. Calculate the next number in the sequence

Thus, we can find the next first difference:

$$ 85 + 108 = 193 $$

Now, add this new difference to the last term of the original sequence:

$$ 140 + 193 = 333 $$