How To Find The First Term Of A Geometric Sequence
Let the showtime term of the sequence be \(a\) and the common ratio exist \(r\).
Utilise the below simulation to calculate the nth term of some geometric progressions.
Input the values of \(northward\) for the number of terms you want to calculate.
Consider a geometric progression whose first term is \(a\) and the common ratio is \(r\).
Can y'all summate the nth term of the geometric progression if the first ii terms are 10 and 20?
Starting time, yous demand to calculate the mutual ratio \(r\) of the geometric series by dividing the second term by the first term.
Then substitute the values of the showtime term \(a\) and the common ratio \(r\) into the formula of the nth term of the geometric progression \(a_{n}=ar^{northward-ane}\).
\[\begin{aligned}a_{due north}&=ar^{n-i}\\&=10(2)^{n-1}\\&=10\frac{2^{n}}{two}\\&=5(2)^{n}\stop{aligned}\]
Important Notes
-
In a geometric progression, each successive term is obtained past multiplying the mutual ratio to its preceding term.
-
The formula for the \(n^{thursday}\) term of a geometric progression whose first term is \(a\) and common ratio is \(r\) is: \(a_n=ar^{n-one}\)
- The sum of n terms in GP whose get-go term is \(a\) and the mutual ratio is \(r\) tin can exist calculated using the formula:
\[S_n=\dfrac{a(one-r^n)}{1-r}\]
Solved Examples
Await at the pattern shown beneath.
Notice that each foursquare is one-half of the size of the square next to information technology.
Which sequence does this design represent?
Solution
Let's write the sequence represented in the effigy.
\[i, \dfrac{1}{two}, \dfrac{1}{4}, \dfrac{1}{viii}, \dfrac{1}{16},...\]
Every successive term is obtained past dividing its preceding term by ii
The sequence exhibits a mutual ratio of \(\dfrac{i}{two}\)
\(\therefore\) The design represents the geometric progression.
Hailey'due south teacher asks her to detect the \( 10^{\text{thursday}}\) term of the sequence: 1, 3, nine, 27, ...
Tin can yous help her?
Solution
Notice that \(\dfrac{iii}{i}=\dfrac{ix}{3}=\dfrac{27}{nine}=3\)
Here, \(r=3\) is a common ratio.
So, the given sequence represents the geometric progression.
The 10th term of the sequence will be given by \(ar^{9}\).
\[\begin{align}ar^{9}&=i\times iii^9\\&=3^9\end{marshal}\]
\(\therefore\) The tenth term of the sequence is \(3^9\)
If the nth term of a GP is 128 and both the first term \(a\) and the mutual ratio \(r\) are 2
Can you calculate the full number of terms in the GP?
Solution
Given,
The nth term of a GP is \(a_n=128\)
The beginning term of the GP is \(a=2\)
The common ratio of the GP is \(r=2\)
At present apply the condition if the first and nth term of a GP are a and b respectively then, \(b=a\cdot r^{n-i}\), to calculate the total number of terms.
\[\begin{align}a_n&=a\cdot r^{n-ane}\\128&=ii\cdot two^{n-i}\\64&=2^{n-ane}\\2^{6}&=ii^{due north-i}\\six&=n-ane\\n&=7\end{marshal}\]
\(\therefore\) There are 7 terms in the GP.
Think Tank
| ane. | There are 25 trees at equal distances of 5 feet in a line with a well, the distance of the well from the nearest tree being 10 feet. A gardener waters all the trees separately starting from the well and he returns to the well later on watering each tree. Notice the total altitude the gardener will cover. |
 |
Interactive Questions
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Let'south Summarize
This mini-lesson targeted the fascinating concept of the nth term of GP. The math journey effectually nth term in GP starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Washed in a style that is non only relatable and like shooting fish in a barrel to grasp only will also stay with them forever. Here lies the magic with Cuemath.
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FAQs
1. What is the full course of GP?
The total form of GP is, "Geometric Progression".
2. What does R equal in geometric progression?
In geometric progression, R is the common ratio of the two consecutive terms.
3. How do yous identify a geometric sequence?
Calculate the ratio of the successive terms of the sequence with the corresponding preceding terms. If all the ratios are equal then the sequence is a geometric sequence.
4. What does the nth term mean?
The nth term represents the general term of the sequence such that \(n=one,2,3,...\) gives the first term, the second term, the third term,... of the sequence.
5. What is the formula for finding the nth term?
The nth term of a geometric sequence with first term \(a\) and the common ratio \(r\) is given by \(a_{north}=ar^{n-1}\).
6. How exercise you lot observe the nth term of a geometric progression with two terms?
First, calculate the common ratio \(r\) by dividing the second term by the offset term. Then use the first term \(a\) and the common ratio \(r\) to calculate the nth term by using the formula \(a_{n}=ar^{n-1}\).
7. What is the difference betwixt arithmetic and geometric progression?
The arithmetic progression has a mutual difference betwixt each consecutive term. While geometric progression has a mutual ratio between each consecutive term.
8. How exercise you solve geometric progression?
To solve the geometric progression offset summate the common ratio \(r\), then use the offset term and the common ratio to calculate the desired terms.
9. What is the formula of geometric progression?
The formula of geometric progression is \(a_{north}=ar^{due north-1}\), where \(a\) i \(r\) are the beginning term and the common ratio respectively.
10. What is the formula for the sum of northward terms in GP?
The sum of geometric progression with beginning term \(a\) and the common ratio \(r\) is given by \(S_n=\dfrac{a(1-r^northward)}{ane-r}\).
Source: https://www.cuemath.com/algebra/nth-term-of-a-gp/
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