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Find the value of ๐ sub two in geometric sequence given six ๐ sub two plus ๐ sub three is equal to 16 ๐ sub one, ๐ sub 10 is equal to 62, and all terms are positive.
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So the first thing to look at is the notation weโve got.
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So weโve got ๐ sub two and ๐ sub three, etc.
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And what this means is the term.
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So, for instance, ๐ sub two is the second term, ๐ sub three will be the third term, etc.
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Okay, weโre also looking at geometric sequence.
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And what a geometric sequence is, is a sequence where there is a common ratio between each term.
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And with the geometric sequence, we have a form to help us find any term within our geometric sequence.
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And that is that ๐ sub ๐ is equal to ๐ sub one multiplied by ๐ to the power of ๐ minus one.
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So what Iโm gonna do is use this in conjunction with the information weโve got to help us solve the problem.
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Well, first of all, weโre gonna take look at the tenth term.
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And that is equal to 62.
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And then if we sub this into our general form, weโre gonna get the tenth term is equal to the first term multiplied by ๐ to the power of 10 minus one.
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So therefore, itโs gonna be the tenth term is equal to the first term multiplied by ๐ to the power of nine, where ๐ is a common ratio.
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As we said, being a geometric sequence is a common ratio between terms.
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So therefore, what we can write is the first term multiplied by the common ratio to the power of nine is equal to 62.
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Okay, great, so thatโs used one piece of information.
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Now, letโs go on and use the other bits of information that we have.
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We were told that six multiplied by the second term plus the third term is equal to 16 multiplied by the first term.
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So letโs first of all find the second and third terms in terms of the first term.
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So our second term is gonna be equal to the first term multiplied by ๐ to the power of two minus one, which is gonna give us a second term is equal to the first term multiplied by the common ratio.
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And the third term is gonna be equal to the first term multiplied by the common ratio squared.
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Ok, great, so howโs this gonna be useful?
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Well, what we can do is we can substitute these in in place of our second and third terms in our equation that weโve got.
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So when we do that, weโre gonna get six ๐ sub one ๐ plus ๐ sub one ๐ squared is equal to 16 ๐ sub one.
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Now, if we take a look at left-hand side, we can see that weโve got a common factor because ๐ sub one or our first term is a common factor in both of our terms.
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So therefore, what we can do is we can rewrite our equation as ๐ sub one multiplied by six ๐ plus ๐ squared is equal to 16๐ sub one.
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Well, therefore, what we can do is divide through by ๐ sub one.
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And when we do that, weโre gonna get six ๐ plus ๐ squared is equal to 16.
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So now, so that I can set up a quadratic equation is equal to zero so that we can solve, what Iโm gonna do is subtract 16 from each side.
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So we get ๐ squared plus six ๐ minus 16 is equal to zero.
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So now, what weโre looking for, because weโre gonna factor our quadratic, is two factors whose sum is positive six and whose product is negative 16.
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Well, our two factors are gonna be positive eight and negative two.
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And thatโs because if we multiply positive eight and negative two, we get negative 16.
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And eight minus two is six.
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Okay, great, so weโve got our factors.
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So now, what we do is we find out what our values of ๐ will gonna be.
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Well, we get ๐ is equal to negative eight or two.
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We got that because to give a value of zero on the right-hand side of the equation, one of our parentheses is gonna have to be equal to zero.
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So if we had ๐ plus eight is equal to zero, then ๐ will be negative eight.
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And if we had ๐ minus two equals zero, then ๐ would have to be equal to two.
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However, are we gonna accept both these values for ๐?
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Well, no, because weโre told that all terms of our geometric sequence are positive.
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So therefore, the common ratio cannot be negative.
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So therefore, we can disregard the common ratio, which is negative eight.
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So we now know that ๐, our common ratio, is equal to two.
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So now, we found our common ratio.
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What weโll look and find is the value of the second term.
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But we canโt quite find it because we still need to find out what the first term is.
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Thatโs ๐ sub one.
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Well, in order to do that, what weโre going to do is substitute in ๐ equals two into the equation that we had for ๐ sub one multiplied by ๐ to the power of nine equals 62.
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We found that because we knew what the tenth term was.
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Well, we now know that the first term multiplied by two to the power of nine is equal to 62.
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So therefore, we know that the first term, ๐ sub one, multiplied by 512 is equal to 62.
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So now, what weโre gonna do is divide through by 512.
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So we get the first term.
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๐ sub one is equal to 62 over 512.
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What Iโm gonna do now is just simplify the fraction on the right-hand side.
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And to do that, what Iโve done is Iโve divided both the numerator and denominator by two because thatโs the factor of both numbers.
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And what I get is ๐ sub one is equal to 31 over 256.
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Okay, great, so we now have the first term.
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We now have the common ratio.
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So what we can do is substitute this in to find the value of this second term.
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So what weโre gonna do, now that we found both the first term and the common ratio, is substitute this in.
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And when we do this, what we get is the second term is equal to 31 over 256 multiplied by two, which is gonna give us a second term is equal to 62 over 256.
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We couldโve just halve the value on the denominator.
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That wouldโve given us the final answer.
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But Iโve done this by multiplying the numerator by two.
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So we get 62 over 256.
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Again, we can simplify this.
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Once again, we can do this by dividing the numerator and denominator by two.
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And once we do this, we get that the second term is equal to 31 over 128.