Video Transcript
Find the sum of the first seven
terms of a geometric sequence given 𝑎 sub five is equal to negative eight
multiplied by 𝑎 sub two and 𝑎 sub four plus 𝑎 sub six is equal to negative
64.
We recall that the 𝑛th term of any
geometric sequence, denoted 𝑎 sub 𝑛, is equal to 𝑎 zero multiplied by 𝑟 to the
power of 𝑛 minus one. 𝑎 sub zero is the first term of
the sequence, and 𝑟 is the common ratio. We are told that the fifth term is
equal to negative eight multiplied by the second term. This means that 𝑎 sub zero
multiplied by 𝑟 to the fourth power is equal to negative eight multiplied by 𝑎 sub
zero 𝑟.
We can divide both sides of this
equation by 𝑎 sub zero. This means that 𝑟 to the fourth
power is equal to negative eight 𝑟. As 𝑟 cannot be equal to zero, we
can divide both sides by 𝑟. Finally, we can cube root both
sides of this equation. This gives us a value of 𝑟 equal
to negative two.
We are also told that the sum of
the fourth and sixth terms is equal to negative 64. This means that 𝑎 sub zero 𝑟
cubed plus 𝑎 sub zero 𝑟 to the fifth power is equal to negative 64. Substituting in 𝑟 equals negative
two gives us negative eight 𝑎 sub zero plus negative 32 𝑎 sub zero is equal to
negative 64. The left-hand side simplifies to
negative 40 𝑎 sub zero. Dividing both sides by negative 40
gives us 𝑎 sub zero is equal to eight-fifths or 1.6.
We now have the first term and
common ratio of our geometric sequence. We recall that the sum of the first
𝑛 terms is equal to 𝑎 sub zero multiplied by one minus 𝑟 to the power of 𝑛
divided by one minus 𝑟. Substituting in our values of 𝑟
and 𝑎 sub zero gives us 𝑆 of seven is equal to 1.6 multiplied by one minus
negative two to the seventh power divided by one minus negative two. Typing this into the calculator, we
get 344 over five. This is equal to 68.8. The sum of the first seven terms of
the geometric sequence is 344 over five or 68.8.
