Find the first three positive terms of a geometric sequence given that their sum is 14 and the sum of their multiplicative inverses is 63 over 32.
We know that any geometric sequence has first term 𝑎 and common ratio 𝑟. This means that the first three terms are 𝑎, 𝑎𝑟, and 𝑎𝑟 squared. We are told that these three terms have a sum of 14. So 𝑎 plus 𝑎𝑟 plus 𝑎𝑟 squared equals 14. If we factor out the common term 𝑎, we are left with 𝑎 multiplied by one plus 𝑟 plus 𝑟 squared is equal to 14. We will call this equation one.
We’re also told in the question that the sum of the multiplicative inverses is 63 over 32. This is another way of saying the reciprocal. The reciprocal of 𝑎 is one over 𝑎. The reciprocal of the second term 𝑎𝑟 is one over 𝑎𝑟. And the reciprocal of the third term is one over 𝑎𝑟 squared. The sum of these is equal to 63 over 32. We can simplify the left-hand side to 𝑟 squared plus 𝑟 plus one over 𝑎𝑟 squared. We do this by multiplying the numerator and denominator of the first fraction by 𝑟 squared and the numerator and denominator of the second fraction by 𝑟.
We can simplify this equation by cross multiplying. This gives us 32 multiplied by 𝑟 squared plus 𝑟 plus one is equal to 63𝑎𝑟 squared. We will call this equation two. By dividing both sides of equation one by 𝑎, we get one plus 𝑟 plus 𝑟 squared is equal to 14 over 𝑎. We can then substitute this into equation two. 32 multiplied by 14 over 𝑎 is equal to 63𝑎𝑟 squared. 32 multiplied by 14 is 448. And when we multiply both sides by 𝑎, we get 448 is equal to 63𝑎 squared 𝑟 squared. We can divide both sides of this equation by 63 so that the left-hand side becomes 64 over nine. We can then square root both sides of this equation such that 𝑎𝑟 becomes eight over three.
Note at this stage that we only want positive terms. 𝑎𝑟 is the second term in any geometric sequence. We will now clear some room to calculate the first and third terms. If 𝑎𝑟 is equal to eight-thirds, then 𝑎 is equal to eight over three 𝑟. We can substitute this into equation one or equation two. Substituting into equation one, we have eight over three 𝑟 multiplied by one plus 𝑟 plus 𝑟 squared is equal to 14. Multiplying both sides by three 𝑟 and distributing the parentheses, we get eight plus eight 𝑟 plus eight 𝑟 squared is equal to 42𝑟. Subtracting 42𝑟 from both sides gives us eight 𝑟 squared minus 34𝑟 plus eight is equal to zero. We can then divide this equation by two, giving us four 𝑟 squared minus 17𝑟 plus four is equal to zero.
This quadratic can be factored into two pairs of parentheses. The first terms are four 𝑟 and 𝑟. The second terms are one and four. As these two numbers need to have a product of four and the middle term is negative, they must both be negative: negative one and negative four. Four 𝑟 squared minus 17𝑟 plus four is equal to four 𝑟 minus one multiplied by 𝑟 minus four.
We can then solve each of these parentheses equal to zero. This gives us two possible values of 𝑟 equal to one-quarter and four. When 𝑟 is equal to one-quarter, 𝑎 is equal to eight divided by three multiplied by one-quarter. This is equal to 32 over three. When 𝑟 is equal to four, 𝑎 is equal to eight divided by three multiplied by four. This is equal to two over three or two-thirds. This means that when 𝑟 is equal to one-quarter, the first three terms are 32 over three, eight over three, and two over three. When 𝑟 is equal to four, the order is reversed. This time, the first term is two-thirds, the second term is eight-thirds, and the third term is thirty-two thirds. These are the first three terms of a geometric sequence where the sum is 14 and the sum of the multiplicative inverses is 63 over 32.