The 6th and 13th term of a geometric progression are 24 and 3/16 respectively.Find the sequence.
clearly, d = (3/16 - 24)/7 = -381/112
Now you can find a, since
a+5d = 24
Now you can list the terms if you want.
it is g.p not a.p!!!!!
To find the common ratio of the geometric progression, we can use the formula:
\[a_n = a_1 \cdot r^{(n-1)}\]
Where \(a_n\) is the \(n\)th term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.
Given that the 6th term is 24, we can substitute the values into the formula:
\[24 = a_1 \cdot r^{(6-1)}\]
\[24 = a_1 \cdot r^5\]
Similarly, for the 13th term, we have:
\[\frac{3}{16} = a_1 \cdot r^{(13-1)}\]
\[\frac{3}{16} = a_1 \cdot r^{12}\]
Now we have a system of equations:
\[24 = a_1 \cdot r^5\]
\[\frac{3}{16} = a_1 \cdot r^{12}\]
To solve this system of equations, we need to eliminate one of the variables. Dividing the two equations, we obtain:
\[\frac{24}{\frac{3}{16}} = \frac{a_1 \cdot r^5}{a_1 \cdot r^{12}}\]
\[8 = r^7\]
Taking the 7th root of both sides, we find:
\[r = \sqrt[7]{8} = 2\]
Substituting this value back into one of the equations, we can solve for \(a_1\). Using the equation:
\[24 = a_1 \cdot r^5\]
\[24 = a_1 \cdot 2^5\]
\[24 = 32a_1\]
\[a_1 = \frac{24}{32} = \frac{3}{4}\]
Now that we have both the common ratio and the first term, we can generate the rest of the sequence.
To find the sequence of a geometric progression, we need to determine the common ratio (r) and the first term (a₁).
Given that the 6th term is 24, we can use the formula for the nth term of a geometric progression:
aₙ = a₁ * r^(n-1)
Substituting n = 6 and a₆ = 24:
24 = a₁ * r^(6-1)
24 = a₁ * r^5
Similarly, for the 13th term being 3/16:
3/16 = a₁ * r^(13-1)
3/16 = a₁ * r^12
Now we have two equations with the same terms (a₁ and r):
1) 24 = a₁ * r^5
2) 3/16 = a₁ * r^12
To solve this system of equations, we can rewrite equation 2) to have the same exponent as equation 1):
3/16 = a₁ * r^(5*2)
3/16 = a₁ * r^10
Now that both equations have the same power of r, we can equate them:
a₁ * r^5 = a₁ * r^10
Since a₁ is common to both sides, we can cancel it out:
r^5 = r^10
To solve for r, we can take the fifth root of both sides:
r^5^(1/5) = r^10^(1/5)
r = r^2
Now we have r = r^2. Since r ≠ 0 (because it's the common ratio), we can divide both sides by r:
1 = r
Therefore, the common ratio (r) is 1.
Now we can substitute this value of r into either of the original equations to solve for a₁. Let's use equation 1):
24 = a₁ * 1^5
24 = a₁
Hence, the first term (a₁) is 24.
Therefore, the sequence of the geometric progression is: 24, 24, 24, 24, 24, 24, ...