We are asked to write sigma notation also referred to as summation notation
for the finite sum. Then determine the sum. Let's first analyze the sum to see
if we can determine a pattern. Notice 7 plus 5 equals 12. 12 plus 5 equals 17. 17
plus 5 equals 22, and this continues all the way out to 62. Because we add 5 to
determine each successive term this is an arithmetic sum or an arithmetic
series which means we can use the arithmetic formulas shown here below to
help us write the sum using sigma notation. Let's begin by determining the
formula a sub n that will generate the terms of the series. The first term in
the series is 7 which means a sub 1 equals 7, and because the pattern is that
we add 5 to determine each successive term the common difference d equals 5.
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 d, we can
use any term in the series and subtract the term before it. Notice 22 minus 17 is
equal to 5, and so with 17 minus 12 as well as 12 minus 7. And now we can use
this information to determine the formula for a sub n. a sub n is equal to
a sub 1 the first term which is 7 plus the quantity n minus 1 times the common
difference d which is 5. And then we will simplify the formula by distributing 5
and combining like terms. We have a sub n equals 7 plus, distributing 5, we have 5
times n which is 5n. And then minus 5 times 1 which is 5. Simplifying, notice 7
minus 5 is equal to 2, giving us a sub n equals 2 plus 5n, which means for sigma
notation or summation notation using the variable n as the index, n begins at the
value of 1. The formula that generates the terms
in the series is two plus 5n. 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 generates the
last term, which is 62. To do this we can set the formula 2 + 5 n equal to 62 and
solve for n. So if we set 2 + 5 n equal to 62 and solve for n, we first subtract
2 on both sides of the equation. Simplifying, we have 5n equals 60. Divide
both sides by 5. Simplifying we have n equals 12. And now we have the sum or the
series written using sigma notation.
Now for the second part of the question we are asked to
determine the partial sum or the sum of the given series. And we can do this
using the formula for S sub n, where n is the number of terms, and therefore this
sum is going to be S sub 12. As soon as we write S sub 12, we know n is equal to
12. So S sub 12 is equal to n divided by 2 which in this case is 12 divided by 2
times the quantity a sub 1 plus a sub n is a sub 1 + a sub 12 or the sum of the
first term in the last term in the series, which in our series is 7 plus 62.
Simplifying, 12 divided by 2 is 6 and 7 plus 62 is 69. 6 times 69 is equal to 414,
which means the sum of the series or S sub 12 is equal to 414. I hope you found
this helpful.