Each term of a sequence is determined by adding half to the proceeding term the sum of the first 25 terms equals to the square of the 25 terms calculate the posible values of the first term
Let the first term of the sequence be x.
According to the given condition, the second term would be x + x/2 = 1.5x
The third term would be 1.5x + 1.5x/2 = 2.25x
And so on,
The 25th term would be 25x
The sum of the first 25 terms of the sequence would be:
x + 1.5x + 2.25x + ... + 25x = x(1 + 1.5 + 2.25 + ... + 25)
This is a sum of an arithmetic series with first term 'a' = 1 and common difference 'd' = 0.5, and 'n' = 25 terms.
The sum of the series is given by the formula:
S = n/2 * (2a + (n-1)d)
S = 25/2 * (2*1 + (25-1)*0.5)
S = 25/2 * (2 + 24*0.5)
S = 25/2 * (14)
S = 175
Given that the sum of the first 25 terms equals the square of the 25 terms:
175 = (25x)^2
175 = 625x^2
x^2 = 175/625
x^2 = 0.28
x = sqrt(0.28) or x = -sqrt(0.28)
Therefore, the possible values of the first term are approximately 0.53 and -0.53.