Question

What are the next two numbers in the pattern: 7,14,28,56?

Answer 1

In this problem we have to find the next two numbers in the given number pattern. Here we can see that the numbers in the pattern in the series seem to be doubled, but we can find the next two numbers in the series by geometric sequence. We know that the formula for the geometric sequence is $y=7\times {{2}^{x-1}}$, we can find the next two numbers using this.

Complete step-by-step solution:
Here we have to find the next two numbers in the pattern: 7,14,28,56
Here the numbers are given in the multiples of seven so its exponential growth
$\Rightarrow y=7\times {{2}^{x-1}}$ ………… (1)
We can see that, in this equation $x$ is the number of the terms in the pattern. For example when $x=1$
$\Rightarrow y=7\times {{2}^{1-1}}$
Anything power of zero is 1. So the answer is 7. Here we got the first term of the pattern. Likewise the first four terms are given in the pattern. So now we have the next two numbers of the pattern that is the fourth and the fifth term. To find the 5th term of the pattern substituting$x=5$ in equation (1) we get,
$\Rightarrow y=7\times {{2}^{5-1}}$
Now simplifying this we get,
$\Rightarrow y=7\times 16$
$\Rightarrow y=112$
Therefore the fifth term of the sequence is 112. Similarly substituting $x=6$ for the 6th term we get,
$\Rightarrow y=7\times {{2}^{6-1}}$
Now simplifying this we get,
$\Rightarrow y=7\times 32$
$\Rightarrow y=224$
Therefore the sixth term of the pattern is 224.
Therefore the solution is 112,224.

Note: This pattern is in the multiples of seven. Where we can get the solution also when they are doubled in the same order. But we have done this problem in geometric sequence. But we must be careful while doing the arithmetic calculation, students may make mistakes while simplifying it.