Certainly! Let's solve the given problems step by step.
### a) Common Ratio
In a geometric progression, each term after the first is obtained by multiplying the previous term by the common ratio [tex]\( r \)[/tex].
Given the first four terms are [tex]\( 3, 6, 12, 24 \ldots \)[/tex]:
To find the common ratio [tex]\( r \)[/tex], we divide the second term by the first term:
[tex]\[ r = \frac{6}{3} = 2 \][/tex]
As well as the third term by the second term:
[tex]\[ r = \frac{12}{6} = 2 \][/tex]
And the fourth term by the third term:
[tex]\[ r = \frac{24}{12} = 2 \][/tex]
Thus, the common ratio [tex]\( r \)[/tex] is [tex]\( 2 \)[/tex].
### b) The 5th Term of the Progression
To find any term in a geometric progression, we use the formula for the [tex]\( n \)[/tex]-th term:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
Where:
- [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the term number.
Given:
- First term [tex]\( a_1 = 3 \)[/tex],
- Common ratio [tex]\( r = 2 \)[/tex],
- We need to find the 5th term ([tex]\( n = 5 \)[/tex]).
Using the formula:
[tex]\[ a_5 = 3 \cdot 2^{(5-1)} = 3 \cdot 2^4 \][/tex]
Calculate [tex]\( 2^4 \)[/tex]:
[tex]\[ 2^4 = 16 \][/tex]
Therefore:
[tex]\[ a_5 = 3 \cdot 16 = 48 \][/tex]
So, the 5th term of the progression is [tex]\( 48 \)[/tex].
### c) The 10th Term of the Progression
Again, we will use the same formula for the [tex]\( n \)[/tex]-th term:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
Given:
- First term [tex]\( a_1 = 3 \)[/tex],
- Common ratio [tex]\( r = 2 \)[/tex],
- We need to find the 10th term ([tex]\( n = 10 \)[/tex]).
Using the formula:
[tex]\[ a_{10} = 3 \cdot 2^{(10-1)} = 3 \cdot 2^9 \][/tex]
Calculate [tex]\( 2^9 \)[/tex]:
[tex]\[ 2^9 = 512 \][/tex]
Therefore:
[tex]\[ a_{10} = 3 \cdot 512 = 1536 \][/tex]
So, the 10th term of the progression is [tex]\( 1536 \)[/tex].
In summary:
a) The common ratio of the progression is [tex]\( 2 \)[/tex].
b) The 5th term of the progression is [tex]\( 48 \)[/tex].
c) The 10th term of the progression is [tex]\( 1536 \)[/tex].
