Please explain a little bit further
4(5^(2n+1))-10(5^(2n-1))/ ( 2(5^n))
= 2(5^(2n-1) ) ( 2(5^2 - 5(1) )/2(5^n))
= 5^(n-1) (50 - 5)
= 225 (5^(n-1) )
To simplify the expression 4(5^(2n+1))-10(5^(2n-1))/ ( 2(5^n)), we can start by applying the exponent rules for multiplication and division.
First, we can simplify the numerator:
4(5^(2n+1))-10(5^(2n-1)) =
= 4 * 5^(2n+1) - 10 * 5^(2n-1)
Next, we can factor out a common term of 2(5^(2n-1)) from the denominator:
2(5^n) = 2 * 5^n
Now, let's combine the numerator and denominator and simplify further:
(4 * 5^(2n+1) - 10 * 5^(2n-1)) / (2 * 5^n)
We can simplify further by factoring out a common term of 2(5^(2n-1)) from the numerator:
= 2(5^(2n-1))(2 * 5^2 - 5 * 1) / (2 * 5^n)
Simplifying the numerator:
= 2(5^(2n-1))(10 - 5) / (2 * 5^n)
= 2(5^(2n-1))(5) / (2 * 5^n)
Next, we can simplify the denominator by canceling out the common factor of 2:
= (5^(2n-1))(5) / (5^n)
Finally, we can simplify the expression by using the exponent rule for division (subtracting the exponents):
= 5^(2n-1 + 1 - n)
= 5^(2n - n - 1 + 1)
= 5^(n-1)(5^2)
= 5^(n-1)(25)
Therefore, the simplified expression is 5^(n-1)(25), which can also be written as 25 * 5^(n-1) or 225(5^(n-1)).