HELP!! Math Probs, Series & Sequences..??
1. Find the terms of the following arithmetic sequence: x, 2x+y, 2x+a, 3x+(1/2)y
2. The sum of three numbers in a geometric series is 52. The 3rd number is 2(1/4) times the sum of the other two. find the numbers in ALL the possible series that result.
I suck at these, any help would be great...thanks in advance!
Comments
(1)Arithmetic sequence: same difference d between terms
d = (term2 - term1) = x + y ---------------- (1)
d = (term3 - term2) = a - y ---------------- (2)
d = (term4 - term2) = x + y/2 - a ---------------- (3)
2d = (term4 - term2) = x - y/2 ---------------- (4)
now (4) =2 * (1) so we have
x - y/2 = 2x + 2y --> x = -5y/2
equating (1) and (2) and substituting for y we have
(-5y/2) + y = a - y ---> a = -y/2
so we have the ratios x : y : a as -5y/2 : y : -y/2
one solution is x=5, y=-2 and a = 1 in which case the series is 5,8,11,14
Another solution is x=-5, y=2, a=-1 in which case the series is -5, -8, -11, -14
Another solution is x=10, y=-4, a = 2: in this case the series is 10,16,22, 28
So there are many solutions with x = 5t, y = -2t, a = t for all values of t ≠ 0
(2) let the geometric sequence of 3 numbers be a, ar, ar² - r is ratio between terms
so a(1 + r + r²) = 52 ------------ (1)
the other given fact No3 =2¼(No1 + No2) translates as
ar² = 9(ar + a)/4 or
4ar² - 9ar - 9a = 0
(4r + 3)(r - 3) = 0 assuming a ≠ 0
so r = 3 and substituting r=3 into (1) we havea(1 + 3 + 9) =52 from which a = 4
and the geometric sequnce is 4, 12, 36
Strawberry Cookie sandwhich is the answer