Respuesta :
a_3 is 45 and a_10 is 98415
Step-by-step explanation:
Given
a_1 = 5
a_2 = 15
First of all we have to calculate the common ratio of the sequence. The common ratio is the ratio between two consecutive terms of a geometric sequence.
So,
[tex]r=\frac{a_2}{a_1} = \frac{15}{5} = 3[/tex]
The explicit formula for geometric sequence is:
[tex]a_n = a_1.r^{n-1}[/tex]
Putting the value of r and a_1
[tex]a_n = 5.(3)^{n-1}[/tex]
Putting n=3 in the explicit formula
[tex]a_3 =5.3^{3-1}\\= 5 * 3^2\\=5 * 9\\= 45[/tex]
Putting n=10 for tenth term
[tex]a_{10} =5.3^{10-1}\\=5*3^9\\=5 * 19683\\=98415[/tex]
Hence,
a_3 is 45 and a_10 is 98415
Keywords: Geometric sequence, Explicit formula
Learn more about geometric sequence at:
- brainly.com/question/10978510
- brainly.com/question/11007026
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The third and tenth term of the considered geometric sequence are evaluated to be: [tex]a_3 = 45, a_{10} = 98415[/tex]
What is a geometric sequence and how to find its nth terms?
There are three parameters which differentiate between which geometric sequence we're talking about.
The first parameter is the initial value of the sequence.
The second parameter is the quantity by which we multiply previous term to get the next term.
The third parameter is the length of the sequence. It can be finite or infinite.
Suppose the initial term of a geometric sequence is [tex]a[/tex] and the term by which we multiply the previous term to get the next term is [tex]d[/tex]
Then the sequence would look like
[tex]a, ad, ad^2, ad^3, ad^4,...[/tex] (till the terms to which it is defined)
Thus, the nth term of such sequence would be [tex]T_n = ad^{n-1}[/tex]
For this case, we're specified that:
[tex]a_1 = 5, a_2 = 15[/tex]
Since we have: [tex]a_2 = a\times d[/tex]
Thus, we get:
[tex]15 = a \times d = 5 \times d\\d = 3[/tex]
Now, the third and tenth terms are obtained as:
[tex]a_3 = a \times d^{3-1} = a \times d^2 = 5 \times 3^3 = 45\\a_3 = 45[/tex]
And,
[tex]a_{10} = a \times d^{10-1} = a \times d^9 = 5 \times 3^9 = 98415\\a_{10} = 98415[/tex]
Therefore, the third and tenth term of the considered geometric sequence are evaluated to be: [tex]a_3 = 45, a_{10} = 98415[/tex]
Learn more about geometric sequence here:
https://brainly.com/question/2735005
