how to find the sum of a geometric series
Sum of Geometric Serial: Definition, Types, Formulas
Sum of Geometric Series: A geometric series is a series where each subsequent number is obtained by multiplying or dividing the number preceding it. The sum of the geometric series refers to the sum of a finite number of terms of the geometric series. A geometric serial can be finite or infinite as there are a countable or uncountable number of terms in the series. The sum of infinite geometric series is greater than the sum of finite geometric series.
Geometric serial have several applications in Physics, Engineering, Biology, Economics, Information science, Queueing Theory, Finance etc. It too has various applications in the field of Mathematics. In this article, we volition provide detailed information on the Sum of Infinite Geometric Series. Coil downwards to learn more well-nigh Sum of Geometric Series!
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Sum of Geometric Serial: Overview
In a Geometric Series, each term is obtained past multiplying or dividing the preceding term by a factor, which is a constant.
In general, nosotros write a geometric sequence in the following format: \(a,\,ar,\,a{r^ii},\,a{r^3},….,\)
Where \(a \to \) is the first term
\(ar \to \) is the second term.
\(r \to \) common ratio (The gene between the terms)
Annotation: In case of a geometric progression, \(r \ne 0,\) because when \(r = 0,\) nosotros get the sequence \(\left\{ {a,\,0,\,0,\,…} \right\}\) which is not geometric.
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Formula for Sum of Infinite Geometric Serial
A geometric series is a set of numbers where each term afterwards the first is found by multiplying or dividing the previous term by a fixed number.
The common ratio, abbreviated every bit \(r,\) is the constant amount.
Permit the commencement, second, 3rd, \(……,\,{n^{{\rm{th}}}}\) term be denoted by \({T_1},\,{T_2},\,{T_3},\,….{T_n},\) and so we tin can write,
\({T_1} = a\)
\({T_2} = ar\)
\({T_3} = a{r^ii}\)
\({T_n} = a{r^{n – one}}\)
\( \Rightarrow r = \frac{{{T_2}}}{{{a_1}}} = \frac{{{T_3}}}{{{T_2}}} = \frac{{{T_4}}}{{{T_3}}}\)
\( \Rightarrow r = \frac{{{T_n}}}{{{T_{n – 1}}}}.\)
Therefore, the mutual ratio can exist adamant by dividing each term by its proceeding term.
Case: Observe the mutual ratio of the geometric series \(1,\,ii,\,four,\,8,\,16,\,….\)
By using the formula,
\( \Rightarrow r = \frac{{{T_2}}}{{{a_1}}} = \frac{{{T_3}}}{{{T_2}}} = \frac{{{T_4}}}{{{T_3}}}\)
\( \Rightarrow r = \frac{2}{ane} = \frac{4}{2} = \frac{8}{four}\)
\( \Rightarrow r = \frac{ii}{ane} = 2\)
And so, the common ratio of the geometric series is \(2.\)
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Sum of northward Terms of a Geometric Series: Types
Sum of northward Terms of a Geometric Series can be classified into two types. Below nosotros have provided the types of Sum of north terms of a Geometric Series:
- Sum of Finite Geometric Series
- Sum of Infinite Geometric Series
1. Sum of Finite Geometric Series
Let us consider that the offset term of a geometric serial is \("a",\) and the common ratio is \(r\) and the number of terms is \(n.\)
There are two cases here.
Case-1: When \(r > ane\)
In this case, the sum of all the terms of the geometric series is given by
\({S_n} = \frac{{a\left( {{r^due north} – 1} \right)}}{{r – i}}.\)
Example-two: When \(r > 1\)
In this case, the sum of all the terms of the geometric serial is given past
\({S_n} = \frac{{a\left( {1 – {r^n}} \right)}}{{1 – r}}.\)
Formula for the Sum of a Finite Geometric Series
Let usa consider that,
\(northward \to \) the number of terms, \(a \to \) first-term
\(r \to \) mutual ratio, \({S_n} \to \) Sum of beginning \(due north\) terms
Permit \({S_n} = a + ar + a{r^2} + ….a{r^{northward – 1}}\)…..(i)
Multiply the equation (i) with \(r,\) we get,
\({S_n} \times r = ar + a{r^two} + a{r^3} + …a{r^{n – 1}} – a{r^northward}\)…… (ii)
Subtract the equation (i) from (ii), we get,
\({S_n}r – {S_n} = a{r^n} – a\)
\( \Rightarrow {S_n}\left( {r – 1} \correct) = a\left( {{r^n} – i} \right)\)
\( \Rightarrow {S_n} = \frac{{a\left( {{r^n} – 1} \right)}}{{\left( {r – 1} \right)}}\)
Hence, the formula to find the sum of the geometric serial is
\({S_n} = \frac{{a\left( {{r^n} – 1} \correct)}}{{\left( {r – one} \correct)}}.\)
two. Sum of Infinite Geometric Serial
In the case where in that location are an space number of terms in the geometric series, meaning in the situation where \(northward \to \infty ,\) the sum of infinite geometric serial is given by
\({S_n} = \frac{a}{{1 – r}},\) where \(a\) is the first term, \(r\) is the common ratio and \(\left| r \right| < one.\)
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Applications of Geometric Series
The geometric series has many applications. Some of them are listed below:
1. Geometric series have huge applications in physics, engineering, biology, economics, reckoner science, queueing theory, finance etc. They are utilised across mathematics.
ii. To calculate the surface area encompassed past a parabola and a straight line, Archimedes utilised the sum of a geometric series.
3. The Koch snowflake'due south interior is made upwardly of an unlimited number of triangles. Geometric series frequently appear in the study of fractals every bit the perimeter, expanse, or book of a self-similar effigy.
4. The knowledge of infinite series makes united states solve aboriginal problems similar zeno's paradoxes.
Solved Examples on Sum of Geometric Series
Q.1. Observe the common ratio of the geometric series \(3,\,6,\,12,\,….\)
Ans: By using the formula, we get,
\(r = \frac{{{T_2}}}{{{a_1}}} = \frac{{{T_3}}}{{{T_2}}}\)
\( \Rightarrow r = \frac{6}{3} = \frac{{12}}{6}\)
\( \Rightarrow r = \frac{ii}{i} = 2\)
Therefore, the mutual ratio of the geometric serial is \(2.\)
Q.2. Find the common ratio of the geometric serial: \(\frac{i}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + ….\)
Ans: By using the formula, we go,
\(r = \frac{{{T_2}}}{{{a_1}}}\)
\( \Rightarrow r = \frac{{\frac{1}{4}}}{{\frac{one}{two}}} = \frac{1}{2}\)
\( \Rightarrow r = \frac{ane}{2}\)
Therefore, the common ratio of the given geometric series is \(\frac{1}{2}.\)
Q.iii. Find the sum of the infinite geometric sequence \(27,\,xviii,\,12,\,8,\,….\)
Ans: From the given geometric series, \(27,\,18,\,12,\,eight,\,….,\) we get,
\(r = \frac{{{a_2}}}{{{a_1}}} = \frac{{eighteen}}{{27}} = \frac{two}{iii}\)
Now, the sum of the infinite geometric series is given past
\(South = \frac{{{a_1}}}{{one – r}}\)
\( \Rightarrow S = \frac{{27}}{{one – \frac{2}{3}}}\)
\( \Rightarrow S = \frac{{27}}{{\frac{{3 – 2}}{3}}} = \frac{{27}}{{\frac{one}{3}}} = 27 \times 3 = 81\)
Hence, the obtained sum is \(81.\)
Q.four. Detect the \({8^{{\rm{thursday}}}}\) term of the following geometric serial: \(4,\,12,\,36,\,108,\,….\)
Ans: From the given geometric series, we get, \(a = 4,\,r = \frac{{12}}{4} = 3,\,n = 8\)
\({T_n} = a{r^{n – 1}}\)
\( \Rightarrow {T_8} = four \times {3^{8 – 1}}\)
\( \Rightarrow {T_8} = four \times {3^7}\)
\( \Rightarrow {T_8} = 4 \times 2187\)
\( \Rightarrow {T_8} = 8748\)
Therefore, the \({8^{{\rm{th}}}}\) term of the given geometric series is \(8748.\)
Q.5. Observe the sum of start \(v\) terms of the geometric series \(10,\,thirty,\,ninety,\,270,\,810,\,2430,\,….\)
Ans: From the given geometric series, we get, \(a = ten,\,r = \frac{{30}}{{10}} = 3\)
In this case \(r > 1.\)
So, the formula to find the sum of the first \(5\) terms of the geometric series is,
\({S_n} = \frac{{a\left( {{r^n} – 1} \correct)}}{{r – 1}}\)
\( \Rightarrow {S_5} = \frac{{x\left( {{3^5} – 1} \correct)}}{{3 – 1}}\)
\( \Rightarrow {S_5} = \frac{{10\left( {243 – 1} \correct)}}{{3 – 1}}\)
\( \Rightarrow {S_5} = \frac{{10\left( {242} \right)}}{two}\)
\( \Rightarrow {S_5} = \frac{{2420}}{2}\)
\( \Rightarrow {S_5} = 1210\)
Therefore, the required sum is \(1210.\)
Q.vi. Find the sum of the first \(seven\) terms of the geometric series if \(a = 1,\,r = 2.\)
Ans: From the given geometric series, we get, \(a = ane,\,r = 2.\)
The formula to notice the sum of the first \(vii\) terms of the geometric serial is,
\({S_n} = \frac{{a\left( {{r^north} – 1} \correct)}}{{r – 1}}\)
\( \Rightarrow {S_7} = \frac{{1\left( {{2^7} – 1} \right)}}{{2 – one}}\)
\( \Rightarrow {S_7} = \frac{{i\left( {128 – 1} \right)}}{1}\)
\( \Rightarrow {S_7} = \frac{{one\left( {127} \right)}}{1}\)
\( \Rightarrow {S_7} = 127\)
Therefore, the sum of the commencement \(seven\) terms of the geometric serial is \(127.\)
Q.vii. Detect the sum, if information technology exists for the geometric series: \(10 + 9 + eight + 7 + ….\)
Ans: In this case, we detect that the given series \(10 + ix + viii + 7 + ….\) is not a geometric series because the ratio between the consecutive terms is non constant.
It is an arithmetics progression. Finding the sum of an space arithmetics series is not possible.
Q.8. Find the sum of first \(5\) terms of the geometric series: \(1,\,ii,\,4,\,eight,\,16,\,32,\,….\)
Ans: From the given geometric series, we get, \(a = one,\,r = \frac{2}{ane} = 2\)
The formula to find the sum of commencement \(5\) terms of the geometric series is,
\({S_n} = \frac{{a\left( {{r^northward} – 1} \correct)}}{{r – one}}\)
Then, \({S_5} = \frac{{1\left( {{ii^v} – one} \right)}}{{ii – 1}}\)
\({S_5} = \frac{{1\left( {32 – ane} \right)}}{1}\)
\({S_5} = \frac{{32}}{one}\)
\({S_5} = 32\)
Therefore, the required sum is \(32.\)
Important Practice Questions on Sum of Geometric Series
Beneath nosotros have provided some of the important practise questions on the sum of geometric serial:
- Find the equivalent fraction of the recurring decimal \(0.797970…..\)
- Discover out the 12th term of the sequence \(3, -half-dozen, nine, -12,….?\)
- Write the outset five terms of a GP whose first term is v and the mutual ratio is 6.
- Discover the sum, if it exists for the geometric serial: \(20 + nineteen + 18 + 17 + ….\)
- Find the sum of the first \(9\) terms of the geometric serial if \(a = 3,\,r = 6.\)
Summary
The geometric progression is a gear up of integers generated past multiplying or dividing each preceding term in such a way that there is a common ratio between the terms (that is not equal to \(0\)), and the sum of all these terms is the sum of the geometric progression. The article includes the definition of geometric series, formula to discover the sum of n terms of finite and infinite geometric series. Nosotros also discussed the derivation of the formula. We mentioned the various applications of the geometric series that helps in understanding its importance. The learning result from the topic is that it will help in understanding the method in finding the sum of the dissimilar sum of geometric series.
FAQs on Sum of Geometric Series
Question 1: Explain the Sum of Geometric Serial with an example?
Reply: A geometric series is a series where each term is obtained by multiplying or dividing the previous term past a constant number, called the common ratio. And, the sum of the geometric series means the sum of a finite number of terms of the geometric series.
Example: Let us consider the series \(27,\,18,\,12,\,…\)
Here, we observe that \(\frac{{18}}{{27}} = \frac{2}{3}\) and \(\frac{{12}}{{xviii}} = \frac{2}{3}.\)
Then, the ratio of the consecutive terms, in this case, is abiding. Hence, the above series can exist chosen a geometric series.
Question 2: How do you observe the sum of a geometric series?
Answer: Allow usa consider that the outset term of a geometric serial is \("a",\) and the common ratio is \(r\) and the number of terms be \(north.\)
To observe the sum of a finite geometric serial, we apply the formula,
\({S_n} = \frac{{a\left( {{r^n} – ane} \right)}}{{r – 1}}\) for \(r > ane\)
\({S_n} = \frac{{a\left( {1 – {r^n}} \correct)}}{{ane – r}}\) for \(r < i.\)
Question iii: What is the sum of \(n\) terms of a Geometric series?
Answer: The formula of the sum of geometric series is given by,
\({S_n} = \frac{{a\left( {{r^due north} – 1} \right)}}{{r – ane}}\) for \(r > 1\)
\({S_n} = \frac{{a\left( {i – {r^due north}} \right)}}{{1 – r}}\) for \(r < 1\)
where \(n \to \)the number of terms, \({a_1} \to \)showtime term and \(r \to \)common ratio.
Question 4: What is the mutual ratio in geometric series?
Answer: The common ratio in the geometric progression is the ratio of any ane term in the series divided by the previous term. The letter \("r"\) is usually used to symbolise it.
Question v: What are the two types of geometric series?
Answer: The two types of geometric serial are listed beneath:
1. Finite geometric series
2. Infinite geometric serial
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