(1) if the second and fourth term of a G.P are 8 and 32 respectively,what is the first term? (i need a practical solution please.)
(2)the eleventh term of an A.P is 25 and its term is 3 find common difference? (also need practical solution.)
(3)given the sequence 0.04 to 25,find the number of terms.(practical solution).
(4)if the 2nd and 5th term of a G.P are 6 and 48 respectively, find the first 4 terms.(practical solution.)
1. 8, and 32 are Base 2 numerals:
8 = 2^3, and 32 = 2^5.
V = 2^(n + 1) = Value of term.
n = term#,
The value of 4th term:
V = 2^(n + 1) = 2^(4 + 1) = 2^5 = 32.
The value of 2nd term:
V = 2^(n + 1) = 2^(2 + 1) = 2^3 = 8.
The Value of 1st term:
V = 2^(n + 1) = 2^(1 + 1) = 2^2 = 4.
1) A practical solution for finding the first term of a geometric progression (G.P) based on the given terms would involve using the formula for the nth term of a G.P.
In this case, the second term (a2) is 8, and the fourth term (a4) is 32. We can use these values to find the common ratio (r) by dividing a4 by a2:
r = a4/a2 = 32/8 = 4
Now, we can use the common ratio to find the first term (a1). Since a2 is the second term, we can divide it by the common ratio to get a1:
a1 = a2/r = 8/4 = 2
Therefore, the first term of the geometric progression is 2.
2) To find the common difference (d) of an arithmetic progression (A.P), we can use the formula for the nth term of an A.P.
Given that the eleventh term (a11) is 25, and the term (n) is 3, we can substitute these values into the formula:
a11 = a1 + (n-1)d
25 = a1 + (3-1)d
25 = a1 + 2d
From this equation, we can deduce that a1 + 2d = 25. Since the term is 3, we substitute n = 3 into the formula:
a3 = a1 + (3-1)d
a3 = a1 + 2d
Now, we can substitute the known values:
25 = a1 + 2d
Since a3 = 25, we equate the two equations:
a1 + 2d = 25
From this equation, we can't determine the exact values of a1 and d without additional information. So, there isn't a unique practical solution to find the common difference without any further details.
3) To find the number of terms in a given sequence, you need the first term, the last term, and the common difference. However, in your question, only the sequence 0.04 to 25 is provided, which is not enough information to determine the number of terms. Therefore, a practical solution cannot be obtained without more details.
4) Given that the second term (a2) is 6 and the fifth term (a5) is 48, we need to find the first four terms of the geometric progression (G.P).
Using the formula for the nth term of a G.P, we can calculate the common ratio (r):
r = (a5/a2)^(1/(5-2)) = (48/6)^(1/3) = 8^(1/3) = 2
Now, we can find the first term (a1) using the common ratio. Since a2 is the second term, we can divide it by the common ratio twice to get a1:
a1 = a2/(r^2) = 6/(2^2) = 6/4 = 1.5
Therefore, the first four terms of the geometric progression are: 1.5, 6, 24, 96.
Sure! I'll provide practical solutions for each of your questions:
(1) For a geometric progression (G.P.), the nth term can be found using the formula: an = a * r^(n-1), where "a" is the first term, "r" is the common ratio, and "n" is the term number.
Given that the second term (a2) is 8 and the fourth term (a4) is 32, we can create two equations using the formula:
a2 = a * r^(2-1) = a * r
a4 = a * r^(4-1) = a * r^3
Dividing these two equations, we get:
a4/a2 = r^3 / r = r^2
Now, substitute the values:
32/8 = r^2
4 = r^2
Taking the square root of both sides, we find:
r = 2
Since we have the value of "r", we can use either a2 = a * r or a4 = a * r^3 to find "a". Let's use a2 = a * r:
8 = a * 2
a = 4
So, the first term (a1) is 4.
(2) For an arithmetic progression (A.P.), the nth term can be found using the formula: an = a + (n-1) * d, where "a" is the first term, "d" is the common difference, and "n" is the term number.
Given that the eleventh term (a11) is 25 and its third term (a3) is 3, we can create two equations using the formula:
a11 = a + (11-1) * d = a + 10d
a3 = a + (3-1) * d = a + 2d
Subtracting these two equations, we get:
a11 - a3 = a + 10d - (a + 2d)
25 - 3 = 8d
22 = 8d
Dividing both sides by 8, we find:
d = 22/8
d = 2.75
So, the common difference (d) is 2.75.
(3) To find the number of terms in a sequence, we can use the formula: n = (last term - first term) / common difference + 1.
Given the sequence 0.04 to 25, we know that the first term (a1) is 0.04 and the last term (an) is 25. However, the common difference is not provided.
If we assume that it is an arithmetic progression, we can find the common difference (d) using the formula from the previous question:
d = (a11 - a3) / (11 - 3) = (25 - 3) / 8 = 22/8 = 2.75
Now, substitute the values into the formula for the number of terms (n):
n = (an - a1) / d + 1 = (25 - 0.04) / 2.75 + 1 = 24.96 / 2.75 + 1 = 9.084 + 1 = 10.084
Rounding this to the nearest whole number, the number of terms (n) is 10.
(4) Using the same formula for a geometric progression (G.P.) as in question 1, we can find the first 4 terms.
Given that the second term (a2) is 6 and the fifth term (a5) is 48, we can create two equations:
a2 = a * r^(2-1) = a * r
a5 = a * r^(5-1) = a * r^4
Dividing these two equations, we get:
a5/a2 = r^4 / r = r^3
Now, substitute the values:
48/6 = r^3
8 = r^3
Taking the cube root of both sides, we find:
r = 2
Using a2 = a * r, we can find the first term (a1):
6 = a * 2
a = 3
So, the first 4 terms of the geometric progression are:
a1 = 3,
a2 = 6,
a3 = a2 * r = 6 * 2 = 12,
a4 = a3 * r = 12 * 2 = 24.
(1) To find the first term of a geometric progression (G.P.) when the second and fourth terms are given, we can use the formula for the nth term of a G.P., which is:
an = a1 * r^(n-1)
Where:
an = nth term
a1 = first term
r = common ratio
Given that the second term (a2) is 8, and the fourth term (a4) is 32, we can solve for the common ratio (r) first.
We have:
a2 = a1 * r^(2-1) --> 8 = a1 * r
a4 = a1 * r^(4-1) --> 32 = a1 * r^3
Divide the second equation by the first equation to eliminate a1:
32 / 8 = (a1 * r^3) / (a1 * r)
4 = r^2
Take the square root of both sides:
r = 2
Now that we have found the common ratio, we can substitute it back into the first equation to solve for the first term:
8 = a1 * 2
a1 = 4
Therefore, the first term of the geometric progression is 4.
(2) To find the common difference of an arithmetic progression (A.P.) when the eleventh term and its corresponding term number are given, we can use the formula for the nth term of an A.P., which is:
an = a1 + (n-1)d
Where:
an = nth term
a1 = first term
d = common difference
Given that the eleventh term (a11) is 25 and its term number (n) is 3, we can substitute these values into the formula:
25 = a1 + (11-1)d --> 25 = a1 + 10d
Since the term number is 3, we can use the equation for the third term to isolate a1:
a3 = a1 + (3-1)d --> a3 = a1 + 2d
Since we know that a3 is 25, we can substitute this in and continue solving:
25 = a1 + 2d
Rearranging the equations, we have a system of equations:
25 = a1 + 10d [Equation 1]
25 = a1 + 2d [Equation 2]
By subtracting Equation 2 from Equation 1, we can eliminate a1:
0 = 8d
d = 0
Therefore, the common difference of the arithmetic progression is 0.
(3) To find the number of terms in a given sequence, we can use the formula for the nth term of an arithmetic sequence, which is:
an = a1 + (n-1)d
Given the sequence 0.04 to 25, we need to determine the value of n when the an is equal to 25. We know the first term (a1) is 0.04, and we need to find the common difference (d).
25 = 0.04 + (n-1)d
Simplifying the equation:
25 - 0.04 = (n-1)d
24.96 = (n-1)d
Since the sequence is given from 0.04 to 25, we can assume that the common difference is positive. If we take d to be positive, then we can solve the equation:
d = 24.96 / (n-1)
We can choose a reasonable value for d, such as 1, and substitute it into the equation:
1 = 24.96 / (n-1)
Cross-multiplying, we get:
n - 1 = 24.96
Adding 1 to both sides:
n = 25.96
Since n represents the number of terms, we can round it down to the nearest whole number because we cannot have a fraction of a term. Therefore, the number of terms in the given sequence is 25.
(4) To find the first four terms of a geometric progression (G.P.) when the second and fifth terms are given, we can still use the formula for the nth term of a G.P. as mentioned in question (1):
an = a1 * r^(n-1)
Given that the second term (a2) is 6, and the fifth term (a5) is 48, we can solve for the common ratio (r) first.
We have:
a2 = a1 * r^(2-1) --> 6 = a1 * r
a5 = a1 * r^(5-1) --> 48 = a1 * r^4
Divide the second equation by the first equation to eliminate a1:
48 / 6 = (a1 * r^4) / (a1 * r)
8 = r^3
Take the cube root of both sides:
r = 2
Now that we have found the common ratio, we can substitute it back into the first equation to solve for the first term:
6 = a1 * 2
a1 = 3
Using the common ratio (r), we can find the first four terms:
- First term (a1) = 3
- Second term (a2) = a1 * r^1 = 3 * 2 = 6
- Third term (a3) = a1 * r^2 = 3 * 2^2 = 12
- Fourth term (a4) = a1 * r^3 = 3 * 2^3 = 24
Therefore, the first four terms of the geometric progression are 3, 6, 12, and 24.