Answer:
a25 = 21
Step-by-step explanation:
In accordance with the given data, this is arithmetic sequence.
If we take that the first term is a1, diference d=1, n-th term is an= a1+ (n-1)d and the sum of the first n terms of the arithmetic sequence is Sn= n/2(a1+an)
Now we don't know the value of the first term but 25-th term is
a25 = a1+(25-1) · 1 = a1+24 => a25=a1+24
The average value of the n-th terms is Sn/n = (n/2(a1+an))/n = 1/2(a1+an)
average value of the 25-th terms is 1/2(a1+a1+24) = 1/2(2a1+24)
That average value is equal to the square of the first term
1/2(2a1+24) = a1∧2 => a1+12 = a1∧2 => a1∧2 - a1 -12 = 0
Now we will factorize this quadratic equation
a1∧2 -4a1 + 3a1 -12 = 0 => (a1∧2 - 4a1) + (3a1 -12) = a1(a1-4) + 3(a1-4) =>
(a1-4) (a1+3) = 0 => a1 - 4 = 0 or a1 + 3 = 0 => a1 = 4 or a1 = -3
because you want the least possible value of the last ( 25-th) term
we choose a1= -3 and finally we get
a25 = a1+24 = -3 + 24 = 21 => a25 = 21
God with you!!!
