Question 863271
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