Express the series eight plus 32
plus 72 plus 128 and so on, all the way up to 512, in sigma notation.
In order to express any series
using sigma notation, we need to write it in the form the sum of 𝑢 sub 𝑟 where 𝑟
takes values from one to 𝑛. We need to work out the expression
for 𝑢 sub 𝑟, which is the general term of the series. At first glance, there doesn’t
appear to be an obvious link between each of the terms in our series. However, they do all have a common
factor of eight. Eight is eight multiplied by one,
32 is eight multiplied by four, 72 is eight multiplied by nine, and so on. One, four, nine, 16, and so on are
the square numbers. This means that the general term of
our series is eight multiplied by 𝑟 squared. As eight squared is equal to 64,
our lower and upper limits are equal to one and eight.
The given series expressed using
sigma notation is the sum of eight 𝑟 squared, where 𝑟 takes values from one to
eight. We could check this answer by
substituting in each of our values of 𝑟 which would obtain the terms eight, 32, 72,
and so on.