In this problem, we want to determine
the sum given in sigma notation.
We have the sum from k equals three to six
of negative five times 1/4 raise to the power
of k minus one.
Looking at the index, remember the number
of terms is equal to the upper limit
minus the lower limit and then plus one,
which means there are only four terms in this series.
So we'll find the sum two ways.
We'll generate the four terms and then find the sum.
But we should also be able to recognize
we do have a geometric series here
where the common ratio r is 1/4,
so we'll also verify we get the same result
by using the partial sum formula
for a geometric series which is shown here below.
So let's first generate the four terms.
The first value of k is three and therefore
the first term in the series is negative five
times 1/4 raise to the power of three minus one
which gives an exponent of two.
And then we have plus, the next value
of k is four, so we have negative five times 1/4
raise to the power of four minus one, which is three.
And then we have plus, the next value
of k is five, giving us negative five times 1/4
raise to the power of five minus one, which is four.
And then finally we have plus, the last value
of k is six, giving us negative five times 1/4
raise to the power of six minus one, which is five.
And now let's simplify.
The first term is negative five times 1/4 squared.
1/4 squared is 1/16 and therefore the first term
is negative 5/16.
And then we have plus negative five times 1/4 cubed.
1/4 cubed is 1/64.
Negative five times 1/64 is negative 5/64.
Plus, for the next term we have negative five
times 1/4 raised to the fourth power.
1/4 to the fourth power is 1 over 256.
Negative five times 1/256 is negative five over 256.
And then for the last term we have negative five
times 1/4 raise to the fifth.
1/4 to the fifth is one over 1,024
and therefore the product is negative five over 1,024.
And now let's go to the calculator and determine the sum.
Let's convert the sum to a fraction
by pressing math, enter, enter.
The sum is negative 425 over 1,024.
But again because we do have a geometric series
let's also use the partial sum formula
for a geometric series shown here to verify
we get the same result.
Where we have the partial sum of the first
n terms of a geometric series is equal
to a sub one times the quantity one minus r
raise to the power of n divided by the quantity one minus r.
Notice the first term of the geometric series
is negative 5/16.
Looking at the formula, we can see the common
ratio r is 1/4.
And the number of terms we can see is four.
Again, to determine the number of terms
in sigma notation we take the upper limit
minus the lower limit and then we add one.
Six minus three is three, plus one is four.
So the sum of the first four terms is equal
to a sub one, which is negative 5/16
times the quantity one minus r
raise to the power of n, which is 1/4,
raise to the fourth power.
All this is divided by one minus r
which is one minus 1/4.
And now let's go to the calculator
and make sure we do get the same result.
But remember, on the calculator we'll have to put
the numerator and denominator in parentheses.
So we have open parentheses, negative five
divided by 16, open parentheses, one minus, open
parentheses, one divided by four,
close parentheses, raise to the fourth,
right arrow, close parentheses for this set
of parentheses, another close parentheses
for the numerator, and then divide it
by open parentheses, one minus one
divided by four, close parentheses, and enter.
And notice how we do get the same decimal
so if we convert to a fraction
by pressing math, enter, enter,
of course we get the same result, negative 425 over 1,024.
So because we only had four terms, the first
method was probably faster.
But if we had a lot more terms, it would probably
be easier to use the partial sum formula.
I hope you found this helpful.