To find the missing number in this sequence, we can look at the relationship between the position of the number and its value.
Each number in the series can be found by taking the square of an even number and then multiplying it by the odd number that sits right in the middle of those squares.
Here is how the pattern works:
First number: You take the square of two, which is four, and then multiply one by one. Or more simply, you can see it as the first odd number (one) multiplied by the square of the first odd number (one).
Second number: You take the square of three, which is nine, and multiply it by one.
Third number: You take the square of five, which is twenty-five, and multiply it by the middle value. A better way to see this specific pattern is using the squares of consecutive odd numbers multiplied by consecutive odd numbers:
One times the square of one equals one.
One times the square of three equals nine.
No, that doesn't quite fit. Let's look at another way.
Let us look at the differences between the numbers:
The difference between one and nine is eight.
The difference between nine and thirty-five is twenty-six.
The difference between thirty-five and ninety-nine is sixty-four.
The difference between ninety-nine and two hundred twenty-five is one hundred twenty-six.
A more elegant way to see this series is by using squares of even numbers and subtracting one:
One: Two squared is four. Four times zero point two five... no.
Let us try multiplying consecutive odd numbers:
One times one is one.
Three times three is nine.
Five times seven is thirty-five.
Nine times eleven is ninety-nine.
Thirteen times seventeen is two hundred twenty-five.
Notice the numbers being multiplied:
The first set is one and one.
The second set is three and three.
The third set is five and seven.
The fourth set is nine and eleven.
The fifth set is thirteen and seventeen.
Looking at the gaps between the numbers we are multiplying:
In the first set, the gap is zero.
In the second set, the gap is zero.
In the third set, the gap is two.
In the fourth set, the gap is two.
In the fifth set, the gap is four.
Following this logic, the next set should have a gap of four.
The next starting odd number in the sequence (one, three, five, nine, thirteen...) follows a pattern where we add two, then two, then four, then four. So, the next starting number after thirteen is seventeen.
The next multiplier will be seventeen plus the gap of four, which is twenty-one.
When you multiply seventeen by twenty-one, the result is three hundred fifty-seven. However, that isn't an option.
Let's look at a simpler pattern:
The first number is one squared times one, which is one.
The second number is three squared times one, which is nine.
The third number is five squared times one point four... no.
Let's try this:
One times one equals one.
Three times three equals nine.
Five times seven equals thirty-five.
Seven times... no.
Let's try:
One times one equals one.
Three times three equals nine.
Five times seven equals thirty-five.
Nine times eleven equals ninety-nine.
Thirteen times seventeen equals two hundred twenty-five.
Wait, let's look at the squares of odd numbers multiplied by the position:
One squared times one is one.
Three squared times one is nine.
Five squared times... no.
Let's look at the squares again:
The square of one is one.
The square of three is nine.
The square of six is thirty-six.
The square of ten is one hundred.
The square of fifteen is two hundred twenty-five.
The numbers being squared are one, three, six, ten, and fifteen.
The difference between these numbers is:
From one to three is two.
From three to six is three.
From six to ten is four.
From ten to fifteen is five.
Following this pattern, the next number to square should be fifteen plus six, which is twenty-one.
The square of twenty-one (twenty-one times twenty-one) is four hundred forty-one.
