1..The first 2 terms of a geometric progression are the same as the first two terms of an arithmetic progression.The first term is 12 and is greater than the second term.The sum of the first 3 terms od the arithmetic progression is 4/3 less than the sum of the first 3 terms of the geometric progression.Determine the value of r,the common ratio of the geometric progression
2..a new sequence is formed by adding together the corresponding terms of a geometric sequence and an arithmetic sequence.The common ratio of the geometric sequence is 2 and the common difference of the arithmetic sequence is 2.The first term of the new sequence is 1,and the second term is 7...
2.1 calculate the third term of the new sequence
2.2write down an expression for the n-th term of the new sequence
a+d = ar
a = 12
r < 1
3/2 (2a+2d) = a(1+r+r^2) - 4/3
3ar = a(1+r+r^2) - 4/3
36r = 12(1+r+r^2) - 4/3
r = 2/3 or 4/3
r=2/3:
12+d = 12* 2/3
d = -4
AP = 12 8 4 0 ...
GP = 12 8 16/3 32/9 ...
check
12+8+4 = 12+8+16/3 - 4/3 YES
4/3
1. Oh boy, we've got ourselves a tricky math problem here! Let's break it down and see what we can do.
We're given that the first two terms of a geometric progression are the same as the first two terms of an arithmetic progression. The first term is 12 and greater than the second term. So, let's call our first term of the arithmetic progression "a" and the common difference "d".
In the geometric progression, our first term is also 12, and the common ratio, let's call it "r". Therefore, the second term of the geometric progression is 12*r.
Now let's consider the sum of the first three terms. In the arithmetic progression, it would be a + (a + d) + (a + 2d). In the geometric progression, it would be 12 + 12*r + 12*r^2.
But wait, there's more! It also says that the sum of the first three terms of the arithmetic progression is 4/3 less than the sum of the first three terms of the geometric progression. So we have:
a + (a + d) + (a + 2d) = 12 + 12*r + 12*r^2 - (4/3)
Now, plug in the values we know:
12 + (12 + d) + (12 + 2d) = 12 + 12*r + 12*r^2 - (4/3)
Simplify that equation and solve for "d" and "r", and you'll have your answer! Good luck, math whiz!
2.1. Now let's move on to the second question, where things are added up in a new sequence. The common ratio of the geometric sequence is 2, and the common difference of the arithmetic sequence is 2. This seems like an interesting mix!
We're given that the first term of the new sequence is 1, and the second term is 7. Let's call our new sequence "S" and the n-th term of the sequence "S_n".
Now, the first term of the geometric sequence is 1 * 2^0 = 1, and the second term is 1 * 2^1 = 2. The first term of the arithmetic sequence is 1, and the second term is 1 + 2 = 3.
To get the first term of the new sequence, we add the corresponding terms of the geometric and arithmetic sequences: 1 + 3 = 4. Hmmm, that's not what we were given. Looks like someone's been playing tricks on us!
But don't be discouraged, my friend! Let's check our calculations and compositions again, because there seems to be a joke hidden in here somewhere.
2.2. Oh, I found it! The joke's on us! It seems the information we were given doesn't lead to a consistent solution. So, I'm afraid there's no expression for the n-th term of the new sequence that we can write down.
But hey, let's not be too serious about this. Sometimes math can be a bit of a funhouse, and we just have to laugh along the way. Keep on chuckling, my friend!
1. Let's denote the common ratio of the geometric progression as r and the common difference of the arithmetic progression as d.
From the given information:
The first term of the geometric progression is 12.
The first term of the arithmetic progression is 12.
The second term of the geometric progression is r * 12.
The second term of the arithmetic progression is 12 + d.
The sum of the first 3 terms of the arithmetic progression is the sum of the first term, second term, and third term. This can be expressed as:
12 + (12 + d) + (12 + 2d)
The sum of the first 3 terms of the geometric progression is:
12 + (r * 12) + (r^2 * 12)
According to the given information, the sum of the first 3 terms of the arithmetic progression is 4/3 less than the sum of the first 3 terms of the geometric progression. This can be represented as an equation:
12 + (12 + d) + (12 + 2d) = (12 + (r * 12) + (r^2 * 12)) - 4/3
Simplifying the equation:
36 + 3d = 12(1 + r + r^2) - 4/3
36 + 3d = 12 + 12r + 12r^2 - 4/3
Multiplying every term by 3 to eliminate the fraction:
108 + 9d = 36 + 36r + 36r^2 - 4
Rearranging terms and simplifying:
9r^2 + 9r - 9d - 76 = 0
2.1. To calculate the third term of the new sequence, we need to find the values of r and d first. Unfortunately, the given information does not provide enough details to determine the values of r and d. Therefore, we cannot calculate the third term of the new sequence.
2.2. To write down an expression for the n-th term of the new sequence, we know that the common ratio for the geometric sequence is 2 (r = 2) and the common difference for the arithmetic sequence is also 2 (d = 2).
The first term of the new sequence is 1, so we can write:
n-th term = 1 + (2^n) + (2 * n)
Therefore, the expression for the n-th term of the new sequence is 1 + 2^n + 2n.
1. To determine the value of the common ratio, r, in the geometric progression, let's first set up the system of equations based on the given information:
Let the first two terms of the arithmetic progression be A and A + B.
Let the first two terms of the geometric progression be 12 and 12 * r.
We know that the sum of the first 3 terms of the arithmetic progression is 4/3 less than the sum of the first 3 terms of the geometric progression. Thus, we have:
A + (A + B) + (A + 2B) = (12 + 12r) + (12r + 12r^2) + (12r^2 + 12r^3) - 4/3
Simplifying the equation, we get:
3A + 3B = 36 + 36r + 36r^2 + 12r^3 - 4/3
Since the first term is greater than the second term, we have:
A > A + B
12 > 12r
Dividing the two equations, we get:
1 > r
So, the value of r, the common ratio of the geometric progression, is less than 1.
2.1. To calculate the third term of the new sequence, we can add the corresponding terms from the geometric and arithmetic sequences.
The geometric sequence has a common ratio of 2, so the terms can be expressed as:
1, 2, 4, 8, ...
The arithmetic sequence has a common difference of 2, so its terms can be expressed as:
1, 3, 5, 7, ...
Adding the corresponding terms:
1 + 2 = 3 (first term)
2 + 4 = 6 (second term)
4 + 5 = 9 (third term)
Therefore, the third term of the new sequence is 9.
2.2. To write down an expression for the n-th term of the new sequence, we can see that each term can be obtained by adding the corresponding terms from the geometric and arithmetic sequences.
The geometric sequence has a common ratio of 2, so its n-th term can be expressed as:
2^(n-1)
The arithmetic sequence has a common difference of 2, so its n-th term can be expressed as:
1 + (n-1) * 2
Therefore, the n-th term of the new sequence can be written as the sum of these two expressions:
2^(n-1) + 1 + (n-1) * 2
This expression gives the n-th term of the new sequence for any positive integer value of n.