Question

Help me i will award medal and fan! The population of a local species of dragonfly can be found using an infinite geometric series where a1 = 42 and the common ratio is three fourths. Write the sum in sigma notation and calculate the sum that will be the upper limit of this population.

Answer 1

the summation of three fourths times 42 to the i minus 1 power, from i equals 1 to infinity.; the sum is divergent.

the summation of three fourths times 42 to the i minus 1 power, from i equals 1 to infinity.; the sum is 168.

the summation of negative 42 times three fourths to the i minus 1 power, from i equals 1 to infinity.; the sum is divergent.

the summation of negative 42 times three fourths to the i minus 1 power, from i equals 1 to infinity.; the sum is 168.

Answer 2

Those are my answer choices

Answer 3

actually hold on... -_-

Answer 4

haha okay and it wants to know if its divergent or the sum is 168

Answer 5

cant believe i have kirbykirby working on my problme xD

Answer 6

Well it will be convergent for sure since the ratio is a fraction less than 1

Answer 7

oh true so that removes two of the answer choices

Answer 8

Actually, before I continue.. does your prof write the geometric series starting from \(a_1\) or \(a_0\)?

Like do they write
\(a_0+a_1+a_2+...\)
or
\(a_1+a_2+...\)

This will make a difference

Answer 9

No i wouldnt think so since i am in highschool, so im pretty sure they dont do that

Answer 10

?

Answer 11

Which one do they use

Answer 12

usually a1

Answer 13

i have never seen a0

Answer 14

ok

Answer 15

The geometric series is written as:
\[\large 42+42\left( \frac{3}{4}\right)+42\left( \frac{3}{4}\right)^2+42\left( \frac{3}{4}\right)^3+...\\
=\large a_1+a_1r+a_1r^2+a_1r^3+...\\
=\large a_1+a_2+a_3+a_4...\]
Now, the geometric series in sigma notation is written as:
\[ \large\sum_{i=0}^{\infty}a_1r^i\]
\[\large \sum_{i=0}^{\infty}42\left(\frac{3}{4} \right)^{i}=\sum_{i=1}^{\infty}42\left( \frac{3}{4}\right)^{i-1} =\large \frac{42}{1-\frac{3}{4}}\]

Answer 16

Thank you so much this was so helpful, you earned yourself a fan and a medal

Answer 17

:) np