Video Transcript
Find the first five terms of the
sequence whose 𝑛th term is given by 𝑎 sub 𝑛 is equal to negative one to the power
of 𝑛 divided by 𝑛 to the fifth power, where 𝑛 is greater than or equal to
one.
In order to work out the first five
terms of a sequence where 𝑛 is greater than or equal to one, we need to substitute
the numbers one, two, three, four, and five into the 𝑛th-term formula. This will give us values for 𝑎 sub
one through 𝑎 sub five. When 𝑛 is equal to one, we have
negative one to the power of one divided by one to the fifth power. When 𝑛 is equal to two, we have
negative one squared over two to the fifth power. When 𝑛 equals three, we have
negative one cubed over three to the fifth power. The fourth and fifth terms, 𝑎 four
and 𝑎 five, are as shown.
When raising negative one to an odd
power, our answer will be negative one. This means that the numerator of
our first, third, and fifth terms will be negative one. Negative one raised to an even
power will give us positive one. This means that the numerator for
our second and fourth term will be positive. One to the fifth power is just
equal to one. Two to the fifth power is 32. Three to the fifth power is 243,
four to the fifth power 1024. And five to the fifth power is
3125. Negative one divided by one is just
negative one. So, 𝑎 sub one is negative one.
None of the other four fractions
can be simplified. Therefore, the first five terms of
the sequence are negative one, one over 32, negative one over 243, one over 1024,
and negative one over 3125.
