S is a geometric sequence. a) Given that (√x - 1), 1, and (√x + 1) are the first three terms of S, find the value of x. b) Show that the 5th term of S is 7 + 5√2
a) We can get the result by calculating the common ratio between elements 1 and 2, and 2 and 3. Rn = 1/√x-1 /Ratio between 1st and 2nd Rn = √x + 1 /Ratio between 2nd and 3rd 1/√x-1 = √x + 1 /*(x + 1) 1 = (√x+ 1)(√x - 1) 1 = x - 1 /Applying A2 - B2= (A + B)(A - B) 2 = x Result: x = 2.b) The nth term can be calculated as an = a1rn-1, where r is the common ratio. We know the ratio r is √2 + 1 from a), n is 5, and a1 is √2 -1, applying result from a) a5= (√2 - 1)(√2 + 1)4 a5= (√2 - 1)(√2 + 1)(√2 + 1)3 /Factoring out (√2 + 1) a5= (2 - 1)(√2 + 1)3 a5= (√2 + 1)3 a5= (√2 + 1)(√2 + 1)2 /Applying (A + B)2= (A2 + 2AB + B2) a5= (√2 + 1)(2 + 2√2 + 1) a5= 2√2 + 4 +√2 + 2 + 2√2 + 1 a5= 7 + 5√2
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