INFINITE GEOMETRIC SERIES

Authored by Patrick Ramp

Mathematics

10th - 12th Grade

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10 questions

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FILL IN THE BLANKS QUESTION

1 min • 1 pt

Find the sum of the infinite geometric series, if it exists. $\sum_{n=1}^{\infty}8\left(\frac{1}{5}\right)^{n-1}$ (a)
If the sum does not exisit, fill boxes with X's

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{9}{1-\left(\frac{1}{5}\right)}=10$

MATH RESPONSE QUESTION

2 mins • 1 pt

Evaluate the infinite geometric series:
1 + 1/5 + 1/25 + ...

$\frac{5}{4}$

Mathematical Equivalence

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{1}{1-\left(\frac{1}{5}\right)}=\frac{5}{4}$

MATH RESPONSE QUESTION

2 mins • 1 pt

Evaluate the infinite geometric series:
3 - 2 + 4/3 - 8/9 + ...

$\frac{9}{5}$

Mathematical Equivalence

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{3}{1-\left(-\frac{2}{3}\right)}=\frac{9}{5}$

FILL IN THE BLANKS QUESTION

1 min • 1 pt

Find the sum of the infinite geometric series, if it exists. $\frac{1}{2}-\frac{5}{3}+\frac{50}{9}-\frac{500}{27}+...$ (a)
If the sum does not exisit, fill boxes with X's

Answer explanation

$\frac{-\frac{5}{3}}{\frac{1}{2}}=-\frac{10}{3}$
r must be between -1 and 1 (and not 0, -1 or 1)

FILL IN THE BLANKS QUESTION

1 min • 1 pt

$\sum_{k=1}^{\infty}\left(\frac{4}{5}\right)^{k-1}$ Find the sum of the series if it exists.
(a)
If the sum does not exisit, fill boxes with X's

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{1}{1-\left(\frac{4}{5}\right)}=5$

FILL IN THE BLANKS QUESTION

1 min • 1 pt

Find the infinite sum of the geometric sequence if its first term is 318 and the common ratio is one-half.

(a)
If the sum does not exisit, fill boxes with X's

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{318}{1-\left(\frac{1}{2}\right)}=636$

FILL IN THE BLANKS QUESTION

2 mins • 1 pt

$1+\frac{1}{4}+\frac{1}{16}\ +\ .\ .\ .\$ Find the sum of the infinite geometric series
Answer as a fraction

(a)

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{1}{1-\left(\frac{1}{4}\right)}=\frac{4}{3}$

FILL IN THE BLANKS QUESTION

1 min • 1 pt

Find the sum of the infinite geometric series, if it exists:
-3 + -3/2 + -3/4 + -3/8...
(a)
If the sum does not exisit, fill boxes with X's

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{-3}{1-\left(\frac{1}{2}\right)}=-6$

FILL IN THE BLANKS QUESTION

1 min • 1 pt

Find the sum of the infinite geometric series:
200 - 100 + 50 - 25 + ...
Answer as a fraction
(a)

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{200}{1-\left(-\frac{1}{2}\right)}=\frac{400}{3}$

10.

FILL IN THE BLANKS QUESTION

1 min • 1 pt

Given the infinite summation, $\sum_{k=1}^{\infty}3\left(\frac{1}{2}\right)^{k-1}$ ,
what is the sum (S)?
(a)
If the sum does not exisit, fill boxes with X's

Answer explanation

$S=\frac{a_1}{1-r}$
$S=\frac{3}{1-\left(\frac{1}{2}\right)}=6$

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INFINITE GEOMETRIC SERIES

Find the sum of the infinite geometric series, if it exists. ∑n=1∞8(15)n−1\sum_{n=1}^{\infty}8\left(\frac{1}{5}\right)^{n-1}n=1∑∞​8(51​)n−1 ​ (a) If the sum does not exisit, fill boxes with X's

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=91−(15)=10S=\frac{9}{1-\left(\frac{1}{5}\right)}=10S=1−(51​)9​=10

Evaluate the infinite geometric series:1 + 1/5 + 1/25 + ...

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=11−(15)=54S=\frac{1}{1-\left(\frac{1}{5}\right)}=\frac{5}{4}S=1−(51​)1​=45​

Evaluate the infinite geometric series:3 - 2 + 4/3 - 8/9 + ...

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=31−(−23)=95S=\frac{3}{1-\left(-\frac{2}{3}\right)}=\frac{9}{5}S=1−(−32​)3​=59​

Find the sum of the infinite geometric series, if it exists. 12−53+509−50027+...\frac{1}{2}-\frac{5}{3}+\frac{50}{9}-\frac{500}{27}+...21​−35​+950​−27500​+... ​ ​ (a) If the sum does not exisit, fill boxes with X's

−5312=−103\frac{-\frac{5}{3}}{\frac{1}{2}}=-\frac{10}{3}21​−35​​=−310​ r must be between -1 and 1 (and not 0, -1 or 1)

∑k=1∞(45)k−1\sum_{k=1}^{\infty}\left(\frac{4}{5}\right)^{k-1}k=1∑∞​(54​)k−1 Find the sum of the series if it exists.​ (a) If the sum does not exisit, fill boxes with X's

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=11−(45)=5S=\frac{1}{1-\left(\frac{4}{5}\right)}=5S=1−(54​)1​=5

Find the infinite sum of the geometric sequence if its first term is 318 and the common ratio is one-half. (a) If the sum does not exisit, fill boxes with X's

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=3181−(12)=636S=\frac{318}{1-\left(\frac{1}{2}\right)}=636S=1−(21​)318​=636

1+14+116 + . . . 1+\frac{1}{4}+\frac{1}{16}\ +\ .\ .\ .\ 1+41​+161​ + . . . Find the sum of the infinite geometric seriesAnswer as a fraction (a) ​

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=11−(14)=43S=\frac{1}{1-\left(\frac{1}{4}\right)}=\frac{4}{3}S=1−(41​)1​=34​

Find the sum of the infinite geometric series, if it exists:-3 + -3/2 + -3/4 + -3/8...​ (a) If the sum does not exisit, fill boxes with X's

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=−31−(12)=−6S=\frac{-3}{1-\left(\frac{1}{2}\right)}=-6S=1−(21​)−3​=−6

Find the sum of the infinite geometric series:200 - 100 + 50 - 25 + ...Answer as a fraction​ (a)

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=2001−(−12)=4003S=\frac{200}{1-\left(-\frac{1}{2}\right)}=\frac{400}{3}S=1−(−21​)200​=3400​

Given the infinite summation, ∑k=1∞3(12)k−1\sum_{k=1}^{\infty}3\left(\frac{1}{2}\right)^{k-1}k=1∑∞​3(21​)k−1 , what is the sum (S)?​ (a) If the sum does not exisit, fill boxes with X's

S=a11−rS=\frac{a_1}{1-r}S=1−ra1​​ S=31−(12)=6S=\frac{3}{1-\left(\frac{1}{2}\right)}=6S=1−(21​)3​=6

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Discover more resources for Mathematics

Find the sum of the infinite geometric series, if it exists. $\sum_{n=1}^{\infty}8\left(\frac{1}{5}\right)^{n-1}$ (a)
If the sum does not exisit, fill boxes with X's

$S=\frac{a_1}{1-r}$
$S=\frac{9}{1-\left(\frac{1}{5}\right)}=10$

Evaluate the infinite geometric series:
1 + 1/5 + 1/25 + ...

$S=\frac{a_1}{1-r}$
$S=\frac{1}{1-\left(\frac{1}{5}\right)}=\frac{5}{4}$

Evaluate the infinite geometric series:
3 - 2 + 4/3 - 8/9 + ...

$S=\frac{a_1}{1-r}$
$S=\frac{3}{1-\left(-\frac{2}{3}\right)}=\frac{9}{5}$

Find the sum of the infinite geometric series, if it exists. $\frac{1}{2}-\frac{5}{3}+\frac{50}{9}-\frac{500}{27}+...$ (a)
If the sum does not exisit, fill boxes with X's

$\frac{-\frac{5}{3}}{\frac{1}{2}}=-\frac{10}{3}$
r must be between -1 and 1 (and not 0, -1 or 1)

$\sum_{k=1}^{\infty}\left(\frac{4}{5}\right)^{k-1}$ Find the sum of the series if it exists.
(a)
If the sum does not exisit, fill boxes with X's

$S=\frac{a_1}{1-r}$
$S=\frac{1}{1-\left(\frac{4}{5}\right)}=5$

Find the infinite sum of the geometric sequence if its first term is 318 and the common ratio is one-half.

(a)
If the sum does not exisit, fill boxes with X's

$S=\frac{a_1}{1-r}$
$S=\frac{318}{1-\left(\frac{1}{2}\right)}=636$

$1+\frac{1}{4}+\frac{1}{16}\ +\ .\ .\ .\$ Find the sum of the infinite geometric series
Answer as a fraction

(a)

$S=\frac{a_1}{1-r}$
$S=\frac{1}{1-\left(\frac{1}{4}\right)}=\frac{4}{3}$

Find the sum of the infinite geometric series, if it exists:
-3 + -3/2 + -3/4 + -3/8...
(a)
If the sum does not exisit, fill boxes with X's

$S=\frac{a_1}{1-r}$
$S=\frac{-3}{1-\left(\frac{1}{2}\right)}=-6$

Find the sum of the infinite geometric series:
200 - 100 + 50 - 25 + ...
Answer as a fraction
(a)

$S=\frac{a_1}{1-r}$
$S=\frac{200}{1-\left(-\frac{1}{2}\right)}=\frac{400}{3}$

Given the infinite summation, $\sum_{k=1}^{\infty}3\left(\frac{1}{2}\right)^{k-1}$ ,
what is the sum (S)?
(a)
If the sum does not exisit, fill boxes with X's

$S=\frac{a_1}{1-r}$
$S=\frac{3}{1-\left(\frac{1}{2}\right)}=6$