Question 863271: Find 'n'
1+3+...+2n-1=10(4n+50)
Do I have to do something with (2n-1+2)(2n-1)
I have no idea. Please help. Found 2 solutions by rothauserc, Theo:Answer by rothauserc(4718) (Show Source):
You can put this solution on YOUR website! The sum of n consecutive odd integers is given by the following formula
Sn = (n/2)(2x1 + 2n - 2) where x1 is the first integer in the sequence
in our example x1 is 1, therefore we have
10(4n+50) = (n/2)(2 +2n -2)
40n+500 = 2n^2 / 2
n^2 -40n -500 = 0
(n-50)(n+10) = 0
n is 50 or -10
in our case n = 50 is the required answer
You can put this solution on YOUR website! The sequence of 1+3+/...+2n-1 = 10(4n+50) is showing you that:
A1 = 1
An = 2n-1
Sn = 10*(4n+50)
An = 2n-1 is the alternate form of finding the terms of an arithmetic sequence.
A1 = 2*1 - 1 = 1
A2 = 2*2 - 1 = 3
A3 = 2*3 - 1 = 5
the common difference is equal to 2.
the primary form of this sequence would therefore be:
An = A1 + (n-1) * d
A1 = 1
d = 2
the formula becomes:
An = 1 + (n-1) * 2
this can also be shown as:
An = 1 + 2 * (n-1)
If you simplify this formula you will get the alternate form.
start with:
An = 1 + 2 * (n-1)
Simplify to get:
An = 1 + 2n - 2
Simplify further to get:
An = 2n - 1
So we are talking about an arithmetic sequence.
The formula for the sum of the terms of an arithmetic sequence is:
Sn = n * (A1 + An) / 2
we know that A1 = 1
we also know that An = 2n-1
we also know that Sn = 10 * (4n + 50)
we substitute in the Sn formula with what we know and solve for what we don't know.
substituting what we know, the formula becomes:
10*(4n + 50) = n * (1 + (2n-1)) / 2
simplify this formula to get:
40n + 500 = (n + 2n^2 - n) / 2
simplify further to get:
40n + 500 = 2n^2 / 2
simplify further to get:
40n + 500 = n^2
subtract 40n and 500 from both sides to get:
n^2 - 40n - 500 = 0
factor this equation to get:
(n-50) * (n+10) = 0
solve for n to get:
n = 50 or n = -10
since n can't be negative, you are left with n = 50
that should be your answer.
to confirm this is a good solution, we'll solve for Sn using the given formula and the formula for the sum of an arithmetic sequence.
the given formula is:
Sn = 10*(4n+50) which becomes:
Sn = 10*(250) which becomes:
Sn = 2500
the formula for the sum of an arithmetic series is:
Sn = n * (A1 + An) / 2
we know that n = 50 and we know that A1 = 1 and we know that An = 2n-1 = 2*50-1 = 99.
the formula becomes:
Sn = 50 * (1 + 99) / 2
simplify this to get:
Sn = 50 * 100/2
simplify further to get:
Sn = 50 * 50
simplify further to get:
Sn = 2500.
both formulas give us the same sum so we're good.
your answer is that n = 50
