We are asked to write sigma notation also referred to as summation notation
for the finite sum. Then determine the sum. Let's first determine the pattern in
the series. Notice 64 minus 8 equals 56. 56 minus 8 equals 48. 48 minus 8 equals
40. Remember subtracting 8 is equivalent to adding negative 8, so because we are
subtracting the same value each time to determine successive terms or we can say
because we are adding negative 8 each time to determine successive terms, this
is an arithmetic series. And therefore we can use the arithmetic formulas shown
here below to help us write these sum using sigma notation. Because the first
term is 64 a sub 1 equals 64. The common difference d is negative 8 because the
pattern is to subtract 8 each time or add negative 8. Notice how there's also a
formula to determine the common difference d given by a sub n minus a
sub n minus 1, which means to determine the common difference, we can take any
term in the series and subtract the term before it. Notice 40 minus 48 is negative
8 and so is 48 minus 56 as well as 56 minus 64. And now we can use a sub 1 and
d to determine the formula a sub n that will generate the terms of the series,
where a sub n is equal to a sub 1 the first term of 64 plus the quantity n
minus 1 times the common difference d, which is negative 8. Now we will simplify
by distributing negative 8 and combining like terms. a sub n equals 64 and now we
will distribute negative 8 so we have plus negative 8 times n which is plus
negative 8n or just minus 8n. And then we have plus negative 8 times negative 1
which is positive 8, giving us plus 8. 64 plus eight equals 72. a sub n equals
72 minus 8n. And now to write the sum using sigma notation we have the
summation and then we'll use the variable n as the index so we begin with
n equals 1. The formula to generate the terms in this series is given by 72
minus 8n. And now we're left to determine the upper limit of the index or the last
value of n which is the value of n that will generate the last term in the
series ,which is 16. To determine this value we will set the formula 72 minus 8n
equal to 16 and solve for n. So doing this we have 72 minus 8n equals 16. Next
step subtract 72 on both sides. Simplifying we have negative 8n equals
16 minus 72 is equal to negative 56. Last step to solve for n is divide both sides by
negative 8. Simplifying we have n equals negative 56 divided by negative 8 is
equal to positive 7. The upper limit of the index or the last value of n is 7.
And now we have the sum written using sigma notation.
The second part of the question asks us to determine the sum of the series. And
because we have an arithmetic series we can use the formula for S sub n to help us
determine the sum. Again because there are seven terms in the series the sum is
equal to S sub seven. As soon as we write s sub 7, we know n is equal to seven so S sub 7
is equal to, n divided by two is seven divided by 2 times the quantity a
sub one plus a sub n is a sub 1 + a sub 7, which is the sum of the first term and
the last term, which in our case is 64 plus sixteen. Simplifying inside the
parentheses 64 plus 16 equals 80. This is equal to seven halves times 80. Let's
write 80 is 80 over one. Before multiplying, we simplify. There's one two
in two and 40 2's in 80, which gives us seven times 40 which equals 280. The sum S sub 7
equals 280. I hope you found this helpful.