Determine the partial sum.
We have the sum of the square
of the quantity two minus 3i
from i equals one to 60.
To determine the partial sum
using the Summation Formulas
shown here on the right,
we first have to square two minus 3i.
Let's go ahead and show the work below.
The square of two minus 3i is equal to
two factors of two minus 3i.
And when multiplying two binomials,
we always have four products.
One, two, three, and four.
Two times two is equal to four,
and then we have two times a negative 3i,
which is equal to negative 6i,
giving us minus 6i,
and then we have negative 3i times two,
which is also negative 6i,
giving us another minus 6i.
And then we have negative 3i
times that negative 3i,
which is positive 9i squared,
giving us plus 9i squared.
Combining like terms,
we have four minus 12i
plus 9i squared,
which means we can write the sum as the sum
of four minus 12i plus 9i squared
from i equals one to 60.
And now let's write this partial sum as a
[mumbles]
difference of three partial sums.
This is equal to the sum of four
from i equals one to 60,
minus the sum of 12i
from i equals one to 60,
and then finally, plus the sum of 9i squared
from i equals one to 60.
And now looking at these Summation Formulas,
we have a formula for the sum of a constant,
the sum of i and the sum of i squared,
which means we'll have to factor out
the 12 here and the nine here.
So this is equal to the sum of four
from i equals one to 60,
then factoring out the 12, we have minus 12
times the sum of i from i equals one to 60.
And then factoring out the nine here,
we have plus nine times the sum of i squared
from i equals one to 60.
And notice for all these sums
and the upper limit of the index is 60.
So applying the Summation Formulas,
the sum of four from i equals one to 60
is equal to c times n, where c is equal to four,
which gives us four times 60
minus 12 times the sum of i
from i equals one to 60,
and the sum of i from i equals one to n
is equal to n times the quantity n
plus one divided by two, in our case n is 60,
so this gives us 12 times 60
times 60 plus one, or 61 divided by two.
And then we have plus nine
times the sum of i squared
from i equals one to 60,
where the sum of i squared is equal to n
times the quantity n plus one
times the quantity 2n plus one divided by six,
which gives us nine times 60,
times n plus one is 61
times 2n plus one, where n is 60,
two times 60 plus one is 121,
and all this is divided by six.
And now we need to simplify.
Four times 60 is equal to 240
minus, simplifying here, 60 divided by two
is equal to 30,
which gives us 12 times 30 times 61,
and then we have plus,
simplifying here, 60 divided by 6 is equal to 10.
So we have nine times 10
times 61 times 121
giving us 240, minus 12 times 30 times 61,
is equal to 21,960,
and then plus nine times 10 times 61 times 121
is equal to 664,290.
And then finally, 240 minus 21,960 plus 664,290
is equal to 642,570.
I hope you found this helpful.