Solve 2 + 6 + 10 + 14 + 18
without actually adding them
square roots
When you have a series of numbers with a common difference, like 2, 6, 10, 14, 18, a shortcut is to use arithmetic series formula to find the sum.
The formula for the sum of an arithmetic series is:
Sum = (n/2)(first term + last term)
where n is the number of terms.
In this case, the first term is 2, the last term is 18, and there are 5 terms.
Sum = (5/2)(2 + 18)
Sum = (5/2)(20)
Sum = 5 * 10
Sum = 50
So, the sum of 2, 6, 10, 14, and 18 is 50.
However, if you want to solve it using square roots, you could try this approach:
Find the square root of each number:
√2 + √6 + √10 + √14 + √18
This would give you the sum of the square roots of the numbers, which is a different calculation.
To solve the given expression without actually adding the numbers, you can use the formula for the sum of an arithmetic series.
The general formula for the sum of an arithmetic series is:
S = (n/2)(2a + (n-1)d),
where:
S is the sum of the series,
n is the number of terms,
a is the first term, and
d is the common difference.
In this case, the series has a common difference of 4, and the first term is 2. We can count the number of terms by dividing the difference between the first and last term by the common difference and adding 1.
The last term can be found by adding the common difference to the first term for every term except the first one, so the last term is 18.
Now, we can calculate the number of terms:
n = (last term - first term) / common difference + 1
= (18 - 2) / 4 + 1
= 16 / 4 + 1
= 4 + 1
= 5.
Using the sum formula, we have:
S = (n/2)(2a + (n-1)d)
= (5/2)(2*2 + (5-1)*4)
= (5/2)(4 + 4*4)
= (5/2)(4 + 16)
= (5/2)(20)
= 100/2
= 50.
Therefore, the sum of the series 2 + 6 + 10 + 14 + 18 is 50.