To solve this problem, we need to find the value of "n" that makes the sum of the series equal to 1456.
The given series is an arithmetic series with a common difference of 2. The sum of an arithmetic series is given by the formula:
S = n/2 * (a + l)
Where:S = Sum of the seriesn = Number of termsa = First terml = Last term
In this case:a = 25l = 2n - 1S = 1456
Plugging in the values, we get:
1456 = n/2 * (25 + 2n - 1)
1456 = n/2 * (24 + 2n)
1456 = n(24 + 2n) / 2
2912 = 24n + 2n^2
Rearranging the equation to form a quadratic equation:
2n^2 + 24n - 2912 = 0
Dividing by 2:
n^2 + 12n - 1456 = 0
Factoring the quadratic equation:
(n + 44)(n - 32) = 0
n = -44 or n = 32
Since the number of terms cannot be negative, n = 32 is the valid solution.
Therefore, the sum of the series 25 + 27 + 29 + ... + (2n - 1) equals 1456 when n = 32.
To solve this problem, we need to find the value of "n" that makes the sum of the series equal to 1456.
The given series is an arithmetic series with a common difference of 2. The sum of an arithmetic series is given by the formula:
S = n/2 * (a + l)
Where:
S = Sum of the series
n = Number of terms
a = First term
l = Last term
In this case:
a = 25
l = 2n - 1
S = 1456
Plugging in the values, we get:
1456 = n/2 * (25 + 2n - 1)
1456 = n/2 * (24 + 2n)
1456 = n(24 + 2n) / 2
2912 = 24n + 2n^2
Rearranging the equation to form a quadratic equation:
2n^2 + 24n - 2912 = 0
Dividing by 2:
n^2 + 12n - 1456 = 0
Factoring the quadratic equation:
(n + 44)(n - 32) = 0
n = -44 or n = 32
Since the number of terms cannot be negative, n = 32 is the valid solution.
Therefore, the sum of the series 25 + 27 + 29 + ... + (2n - 1) equals 1456 when n = 32.