Video Transcript
Evaluate the sum of one-quarter
multiplied by two to the power of 𝑟 minus one where 𝑟 takes values from four to
nine.
The values four and nine are the
lower and upper limits of 𝑟, respectively. The Greek letter 𝛴 means the sum
of. In this question, we need to find
the sum of six terms when 𝑟 equals four, five, six, seven, eight, and nine. If we look at our expression, we
notice that one-quarter is a constant. This means that we can rewrite our
expression as shown: one-quarter multiplied by the sum of two to the power of 𝑟
minus one where 𝑟 takes values from four to nine. We now need to substitute each of
the values of 𝑟 into our expression. Four minus one is equal to
three. So we have two cubed. When 𝑟 is equal to five, we have
two to the fourth power. Repeating this process, we have the
six terms as shown.
We need to find the sum of these
and then multiply our answer by one-quarter. Two cubed is equal to eight, two to
the fourth power is equal to 16, and so on. As the six numbers inside our
brackets sum to 504, we need to find one-quarter of 504. This is equal to 126. The sum of one-quarter multiplied
by two to the power of 𝑟 minus one where 𝑟 takes values from four to nine is
126.
