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Question 765337: The first term of geometric sequence is a^-4, the second term is a^x. If the 8th term is a^52, what is the value of x?
Please advise me how to solve this,
I figured out a^x= ar and ar^7 = 52, and I don't know where it goes from here.
Thank you
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! a.1 = the first term in the sequence.
a.2 = second term in the sequence.
a.n = nth term in the sequence.
you are given that a.1 = a^-4 (first term in the sequence).
you are also given that a.2 = a^x (second term in the sequence).
you are also given that a.8 = a^52 (eighth term in the sequence).
first thing you want to do is solve for r.
you can use the first and second terms in the sequence to do this.
first term is a^-4.
second term is a^x
formula for second term is:
a.2 = a.1 * r^(n-1)
in the second term of the sequence, n is equal to 2.
in the first term of the sequence, n is equal to 1.
you get:
a.2 = a.1 * r^(2-1) which becomes a.2 = a.1 * r^1 which becomes a.2 = a.1 * r
your formula becomes:
a.2 = a.1 * r (second term in the sequence is equal to a.1 * r).
since a.2 is equal to a^x and a.1 is equal to a^-4, this formula becomes:
a^x = a^-4 * r
you can now solve for r in terms of a as follows:
divide both sides of this equation by a^-4 to get:
r = a^x / a^-4
the laws of exponent arithmetic make this equal to:
r = a^(x-(-4)) which becomes:
r = a^(x+4)
you now have the value of r.
r is equal to a^(x+4)
you are given that the 8th term in the sequence is equal to a^52.
the 8th term in the sequence is given by the formula:
a.8 = a.1 * r^(8-1) which becomes:
a.8 = a.1 * r^7)
you know that a.1 is equal to a^-4 and you know that r = a^(x+4) and you know that a.8 is equal to a^52, so you can susbtitute in the formula for a.8 to get:
a^52 = a^-4 * (a^(x+4))^7
by the rules of exponent arithmetic, a^(x+4))^7 becomes a^(7*(x+4)) which becomes
a^(7x+28).
your formula becomes:
a^52 = a^-4 * a^(7x+28).
by the rules of exponent arithmetic, a^-4 * a^(7x+28) is equal to a^(-4 + 7x + 28) which is equal to a^(7x + 24).
your formula becomes:
a^52 = a^(7x + 24)
this can only be true if 52 = 7x + 24.
now you can solve for x as follows:
subtract 28 from both sides of that equation to get:
52 - 24 = 7x
simplify to get:
28 = 7x
solve for x to get:
x = 4
that's your answer.
x = 4
you can confirm your answer is good by substituting for x in all the equations that use it.
before you do that, however, solve for r with x = 4.
when x = 4, r = a^(x+4) becomes a^8.
the first term in your sequence is equal to a^-4 (given).
the second term in your sequence is equal to a^x which is equal to a^4 because x = 4.
since a.2 = a.1 * r, this means that a.2 = a.1 * a^8 because r = a^8.
since a.2 = a^4 and a.1 = a^-4, then this formula becomes:
a^4 = a^-4 * a^8.
by the rules of exponent arithmetic, a^-4 * a^8 = a^(-4 + 8) = a^4 and you get:
a^4 = a^4.
by similar manipulations you can solve for a.8 as well.
a.8 = a.1 * r^(8-1) which is equal to a.1 * r^7.
since r = a^8, this formula becomes:
a.8 = a.1 * (a^8)^7 which becomes:
a.8 = a.1 * a^56
since a.1 = a^-4 and a^8 = a^52, this formula becomes:
a^52 = a^-4 * a^56
by the rules of exponent arithmetic, a^-4 * a^56 = a^(-4 + 56) which becomes a^-52.
your formula becomes:
a^52 = a^52 which is true, confirming the value of x = 4 is good.,
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