Find the sum of the series 4+44+444+....to n terms.
To find the sum of the series 4 + 44 + 444 + ... to n terms, we can create a pattern and then use a formula to calculate the sum.
If we look closely at the numbers in the series, we can observe that each term is formed by "n" number of 4's followed by a 4 at the end.
Let's take a few terms of the series to understand the pattern:
n = 1: 4
n = 2: 44
n = 3: 444
n = 4: 4444
From the pattern, we can see that each term can be calculated as (4 * 10^(n-1)) + 4.
To find the sum of the series, we need to calculate the sum of all these terms. Let's call it S.
S = (4 * 10^(0)) + (4 * 10^(1)) + (4 * 10^(2)) + ... + (4 * 10^(n-2)) + (4 * 10^(n-1))
Now, let's factor out the common term 4:
S = 4 * (10^(0) + 10^(1) + 10^(2) + ... + 10^(n-2) + 10^(n-1))
Using the formula for the sum of a geometric series, which is given by:
Sum = (a * (r^n - 1)) / (r - 1)
where "a" is the first term, "r" is the common ratio, and "n" is the number of terms, we can see that a = 1, r = 10, and n = n.
Now substituting these values into the formula, we get:
S = 4 * ((10^n - 1) / (10 - 1))
Simplifying further, the sum of the series 4 + 44 + 444 + ... to n terms is:
S = (4 * (10^n - 1)) / 9
To find the sum with a given value of n, simply plug in the value of n into the formula:
Sum = (4 * (10^n - 1)) / 9