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Question 872865: In a progression, the third term has a value of 7 and the tenth term has a value of 17.5. Determine the values of the first term; the common difference if it is an AP and the common ratio if it is a GP.
Also, find the sum of the first two terms in each case.
Please help me, I've been struggling on this question for hours!
Thank you so much!
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! 1)AP is defined for the nth term as Xn = a + d(n-1), a is first term and d is common difference. We are given
7 = a + d(2)
17.5 = a + d(9)
solve first equation for a
a = 7 - 2d
substitute for a in second equation
17.5 = 7 - 2d + 9d
7d = 10.5
d = 10.5 / 7 = 3/2
a = 7 - 2(3/2) = 4
For AP a = 4, d = 3/2
We are asked for the sum of the first two terms, X1 and X2
X1 = 4 + (3/2)(1-1) = 4
X2 = 4 + (3/2)(2-1) = 5.5
X1 + X2 = 9.5
2)GP is defined for the nth term as Xn = ar^(n-1), where a is first term and r is the common ratio. We are given
7 = ar^2
17.5 = ar^9
solve first equation for a
a = 7/r^2
substitute for a in second equation
17.5 = (7/r^2)r^9
17.5 = 7r^7
r^7 = (17.5/7)
r^7 = 2.5
r = 1.139852281
a = 7 / (1.139852281)^2 = 5.387668856
For GP a = 5.387668856, r = 1.139852281
We are asked for the sum of X1 and X2
X1 = (5.387668856)(1.139852281)^(1-1) = 5.387668856
X2 = (5.387668856)(1.139852281)^(2-1) = 6.141146635
X1 + X2 = 5.387668856 + 6.141146635 = 11.528815491
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