LESSON 1.3 - ACTIVITY 5
Find the sum of the following infinite geometric sequence.
plsss yung tama,malapit napo deadline namin sana naman yung tamang answer
Find the sum of the following infinite geometric sequence.
plsss yung tama,malapit napo deadline namin sana naman yung tamang answer
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Verified answer
✏️INFINITE SERIES
#1:
» Determine the common ratio.
» Find the infinite sum of the given sequence.
#2:
» Determine the common ratio.
» Find the infinite sum of the given sequence.
#3:
» Determine the common ratio.
» Find the infinite sum of the given sequence.
#4:
» Determine the common ratio.
» Find the infinite sum of the given sequence.
#5:
» Determine the common ratio.
» Find the infinite sum of the given sequence.
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Answer:
✏️INFINITE SERIES
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\underline{\mathbb{DIRECTIONS}:}
DIRECTIONS:
Find the sum of the following infinite geometric sequence.
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\underline{\mathbb{ANSWERS}:}
ANSWERS:
\qquad\Large 1) \tt\: \green{S_\infin = 14}1)S
∞
=14
\qquad\Large 2) \tt\: \green{S_\infin = \frac{16}{3}}2)S
∞
=
3
16
\qquad\Large 3) \tt\: \green{S_\infin = 10}3)S
∞
=10
\qquad\Large 4) \tt\: \green{S_\infin = 8}4)S
∞
=8
\qquad\Large 5) \tt\: \green{S_\infin = 81}5)S
∞
=81
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\underline{\mathbb{SOLUTIONS}:}
SOLUTIONS:
#1:
» Determine the common ratio.
\begin{gathered} \begin{aligned} & \bold{Formula:} \\ & \boxed{\rm r = \frac{a_n}{a_{n-1}}} \end{aligned} \end{gathered}
Formula:
r=
a
n−1
a
n
\begin{gathered} \rm r = \frac{a_2}{a_1} = \frac{\,7\,}{2} \div 7 = \frac{\,1\,}{2} \\ \end{gathered}
r=
a
1
a
2
=
2
7
÷7=
2
1
\begin{gathered} \rm r = \frac{a_3}{a_2} = \frac{\,7\,}{4} \div \frac{\,7\,}{2} = \frac{\,1\,}{2} \\ \end{gathered}
r=
a
2
a
3
=
4
7
÷
2
7
=
2
1
» Find the infinite sum of the given sequence.
\begin{gathered} \begin{aligned} & \bold{Formula:} \\ & \boxed{\rm S_\infin = \frac{a_1}{1 - r}} \end{aligned} \end{gathered}
Formula:
S
∞
=
1−r
a
1
\begin{gathered} \rm S_\infin = \frac{7}{1 - \frac{\,1\,}{2}} \\ \end{gathered}
S
∞
=
1−
2
1
7
\begin{gathered} \rm S_\infin = \frac{\,7\,}{\frac{\,1\,}{2}} \\ \end{gathered}
S
∞
=
2
1
7
\begin{gathered} \rm S_\infin = 7 \div \frac{\,1\,}{2} \\ \end{gathered}
S
∞
=7÷
2
1
\begin{gathered} \rm S_\infin = 7 \cdot 2 \\ \end{gathered}
S
∞
=7⋅2
\rm S_\infin = 14S
∞
=14
\therefore∴ The sum of the given infinite geometric sequence is 14.
\:
#2:
» Determine the common ratio.
\begin{gathered} \rm r = \frac{a_2}{a_1} = \frac{\,1\,}{4} \\ \end{gathered}
r=
a
1
a
2
=
4
1
\begin{gathered} \rm r = \frac{a_3}{a_2} = \frac{\,1\,}{4} \div 1 = \frac{\,1\,}{4} \\ \end{gathered}
r=
a
2
a
3
=
4
1
÷1=
4
1
» Find the infinite sum of the given sequence.
\begin{gathered} \rm S_\infin = \frac{4}{1 - \frac{\,1\,}{4}} \\ \end{gathered}
S
∞
=
1−
4
1
4
\begin{gathered} \rm S_\infin = \frac{\,4\,}{\frac{\,3\,}{4}} \\ \end{gathered}
S
∞
=
4
3
4
\begin{gathered} \rm S_\infin = 4 \div \frac{\,3\,}{4} \\ \end{gathered}
S
∞
=4÷
4
3
\begin{gathered} \rm S_\infin = 4 \cdot \frac{\,4\,}{3} \\ \end{gathered}
S
∞
=4⋅
3
4
\begin{gathered} \rm S_\infin = \frac{16}{3} \\ \end{gathered}
S
∞
=
3
16
\therefore∴ The sum of the given infinite geometric sequence is 16/3.
\:
#3:
» Determine the common ratio.
\begin{gathered} \rm r = \frac{a_2}{a_1} = \frac{2.5}{5} = 0.5 \\ \end{gathered}
r=
a
1
a
2
=
5
2.5
=0.5
\begin{gathered} \rm r = \frac{a_3}{a_2} = \frac{1.25}{1.5} = 0.5 \\ \end{gathered}
r=
a
2
a
3
=
1.5
1.25
=0.5
» Find the infinite sum of the given sequence.
\begin{gathered} \rm S_\infin = \frac{5}{1 - 0.5} \\ \end{gathered}
S
∞
=
1−0.5
5
\begin{gathered} \rm S_\infin = \frac{5}{0.5} \\ \end{gathered}
S
∞
=
0.5
5
\rm S_\infin = 10S
∞
=10
\therefore∴ The sum of the given infinite geometric sequence is 10.
\:
#4:
» Determine the common ratio.
\begin{gathered} \rm r = \frac{a_2}{a_1} = \frac{\,2\,}{4} = \frac{\,1\,}{2} \\ \end{gathered}
r=
a
1
a
2
=
4
2
=
2
1
\begin{gathered} \rm r = \frac{a_3}{a_2} = \frac{\,1\,}{2} \\ \end{gathered}
r=
a
2
a
3
=
2
1
» Find the infinite sum of the given sequence.
\begin{gathered} \rm S_\infin = \frac{4}{1 - \frac{1}{2}} \\ \end{gathered}
S
∞
=
1−
2
1
4
\begin{gathered} \rm S_\infin = \frac{\,4\,}{\frac{1}{2}} \\ \end{gathered}
S
∞
=
2
1
4
\begin{gathered} \rm S_\infin = 4 \div \frac{\,1\,}{2} \\ \end{gathered}
S
∞
=4÷
2
1
\begin{gathered} \rm S_\infin = 4 \cdot 2 \\ \end{gathered}
S
∞
=4⋅2
\begin{gathered} \rm S_\infin = 8 \\ \end{gathered}
S
∞
=8
\therefore∴ The sum of the given infinite geometric sequence is 8.
\:
#5:
» Determine the common ratio.
\begin{gathered} \rm r = \frac{a_2}{a_1} = \frac{18}{54} = \frac{\,1\,}{3} \\ \end{gathered}
r=
a
1
a
2
=
54
18
=
3
1
\begin{gathered} \rm r = \frac{a_3}{a_2} = \frac{6}{18} = \frac{\,1\,}{3} \\ \end{gathered}
r=
a
2
a
3
=
18
6
=
3
1
» Find the infinite sum of the given sequence.
\begin{gathered} \rm S_\infin = \frac{54}{1 - \frac{\,1\,}{3}} \\ \end{gathered}
S
∞
=
1−
3
1
54
\begin{gathered} \rm S_\infin = \frac{54}{\frac{\,2\,}{3}} \\ \end{gathered}
S
∞
=
3
2
54
\begin{gathered} \rm S_\infin = 54 \div \frac{\,2\,}{3} \\ \end{gathered}
S
∞
=54÷
3
2
\begin{gathered} \rm S_\infin = 54 \cdot \frac{\,3\,}{2} \\ \end{gathered}
S
∞
=54⋅
2
3
\begin{gathered} \rm S_\infin = \frac{162}{2} \\ \end{gathered}
S
∞
=
2
162
\rm S_\infin = 81S
∞
=81
\therefore∴ The sum of the given infinite geometric sequence is 81.
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