Write sigma notation or summation notation for the finite sum. Then
determine the sum. Let's first analyze the sum to see if we notice a pattern.
Notice 4 times 3 is equal to 12. 12 times 3 equals 36.
36 times 3 equals 108, and the pattern continues all the way out to 2916.
Because we are multiplying by 3 each time to determine each successive term,
this is called a geometric series, and we can use the formulas for geometric
sequences or geometric series here to help us write the sum using sigma
notation. Because the first term is 4 a sub 1 is equal to 4. Because the pattern
is to multiplied by 3 each time, the common ratio r is equal to 3. There's
also a formula to determine the common ratio r given by a sub n divided by a
sub n minus 1, which means we can determine r by selecting any term in the
series and dividing by the term before it. Notice 12 divided by 4 equals 3, and
so is 36 divided by 12 and 108 divided by 36. Now that we have a sub 1 and r, we
can determine a sub n which is the formula that will generate the terms of
a series or of the sum. For a geometric series, a sub n is equal to a sub 1 which
is 4 times r raised to the power of n minus 1, which in our case is 3 raised to
the power of n minus 1. And now let's begin to write the sum using sigma notation. We
have the summation because the variable is n we have the index starting at n
equals 1. The formula to generate the terms is 4 times 3 raised to the power
of n minus 1. And now we need to find the upper limit of the index or the last
value of n, which is a value of n that will generate the last term which is 2916.
To do this, we will set the formula a sub n equal to 2916 and solve for n. This
gives us the equation 4 times 3 raised to the power of n minus 1 equals
2916. For next step, we will divide both sides by four. Simplifying this gives us
three raised to the power of n minus 1 equals 2916 divided by 4 equals 729.
And now to solve the equation. We could use logarithms but let's
go ahead and write 729 as 3 raised to a power. To do this we will determine the
prime factorization of 729. 729 is equal to 3 times 243, where 3 is prime. 243 is
equal to 3 times 81. 3 is prime. 81 is equal to 9 times 9, and 9 is equal to 3
times 3 here as well as here. Notice 729 is equal to 6 factors of 3 or 3 to the 6th.
This gives us the equation 3 raised to the power of n minus 1 equals 3 to the 6th.
Because these are equal and the bases are the same, the exponents must be equal,
which means n minus 1 must equal 6. And therefore n is equal to 7. 7 is the upper
limit of the index or the last value of n. And now we have the sum written using
sigma notation. The last step is determine the sum, which for a geometric
series is equal to S sub n given by this formula here. Because the index goes from
n equals 1 to 7, we know we have seven terms in the sum or in the series. And
therefore the sum is equal to S sub 7. For a geometric series, this is equal to
a sub 1 which is the first term which is 4 times the fraction where the numerator
is 1 minus r raised to power of n which is 1 minus 3 raised to the power of 7,
divided by the quantity 1 minus r which is 1 minus 3. And now we need to simplify.
Let's simplify the numerator and denominator of the fraction first. 3 to
the 7th is equal to 2187, which means S sub 4 is equal to 4 times 1 minus 2187
divided by 1 minus 3 is negative 2. 1 minus 2187 is equal to negative 2186.
This gives us 4 times negative 2186 divided by negative 2. And negative
2186 divided by negative 2 equals 1093, giving us S sub 7 equals 4 times
1093. And 4 times 1093 equals 4372,
which means the sum S sub 7 equals 4372.
I hope you found this helpful.