We're asked to determine
the sum given in sigma notation.
We have the sum from n equals 21 to 60 of two plus three n.
We should be able to recognize we have
an arithmetic series here, because the formula is
two plus three n, where the common difference, d, is three.
Let's go ahead and generate
some of the terms of this series.
Notice how the first value of n is 21,
and therefore the first term is two plus three times 21.
The next value of n is 22, and therefore the next term
is going to be two plus three times 22.
The next value of n is 23, and therefore the next term
is two plus three times 23.
Of course this continues.
Let's go ahead and find the last term of the series,
when n is equal to 60, and therefore
the last term is two plus three times 60.
Now let's simplify.
Three times 21 is 63, two plus 63 is 65,
plus next we have three times 22 which is 66,
two plus 66 is 68, plus next we have
three times 23 which is 69, two plus 69 is 71,
plus, and then the last term, we have
three times 60, which is 180, two plus 180 is 182.
If we didn't recognize it before, we should now recognize
we do have an arithmetic series,
because the terms are increasing by three.
Let's make a note over here the first term,
a sub one, is 65, and the common difference,
d, is positive three.
Remember if we needed to, we could determine
the common difference, d, by using the formula
d equals a sub n minus a sub minus one,
which means we can find d by selecting any term
in the series, except the first term,
and subtracting the term before it.
Notice 68 minus 65 is three, 71 minus 68 is also three.
We know the last term in the series is 182,
and therefore a sub n is equal to 182.
Now we need to determine n,
the number of terms in the series.
Notice how the index goes from 21 to 60,
so you might be thinking we should take
the upper limit of 60 and subtract the lower limit of 21
to determine the number of terms.
60 minus 21 is 39, which is not the number of terms.
There's actually one more term, or 40 terms.
To better understand this, let's look at a simpler example.
Let's say we had the sum from n equals one to four of n.
If we expand this, we have
one plus two plus three plus four.
We can easily see we have four terms,
but if we take the upper limit and subtract the lower limit,
we have four minus one, which is three.
Again, we have four terms, so we do have to add one
to get the correct number of terms.
Now we have all the information we need
in order to determine the sum of the arithmetic series
using the formula S sub n equals n divided by two
times the quantity a sub one plus a sub n.
The partial sum of the first 40 terms
of the arithmetic series is equal to
n divided by two, which is 40 divided by two,
times the quantity a sub one plus a sub n,
where a sub one is 65 and a sub n,
which is really a sub 40, is 182.
We have 65 plus 182.
Simplifying, we have S sub 40 is equal to
40 divided by 2 is 20.
65 plus 182 is 247.
20 times 247 is equal to 4940.
This is the sum of the series.
I hope you found this helpful.