The sum of three different numbers which are consecutive terms of a geometric progression is 28. The numbers are also the first, third and seventh term of an arithmetic progression. Find the three numbers
Use your definitions.
a + ar + ar^2 = 28
a(1+r+r^2) = 28
for the AP information:
the first terms match ----> a=a
the 2nd of the GP = 3rd of AP
ar = a+2d
ar-a = 2d
a(r-1) = 2d **
the 3rd of the GP = 7th of the AP
ar^2 = a + 6d
ar^2 - a = 6d
a(r^2 - 1) = 6d ***
divide *** by **
a(r^1 - 1) / a(r - 1) = 6d / 2d
r+1 = 3
r = 2
then in a(1+r+r^2) = 28
a(1+2+4) = 28
7a = 28
a = 4
now in **
a(r-1) = 2d
4(1) = 2d
d = 2
so the GP is 4, 8, 16, ...
the AP is 4, 6, 8, 10, 12, 14, 16, ...,
notice the first terms match,
the 2nd of the GP is the 3rd of the AP
the 3rd of the GP is the 7th of the AP
I know this is a long solution, and I am sure there is a shorter version, but
I just followed by train of thought, and used your basic definitions.
Not sure
Please answer it quickly it is argent
To solve this problem, let's assign variables to the numbers in the geometric progression. Let's call the first term "a" and the common ratio "r".
Since the sum of three consecutive terms in a geometric progression is given as "28", we can write the equation:
a + ar + ar^2 = 28
Now let's look at the arithmetic progression. The first term of the arithmetic progression is the first term (a) of the geometric progression. The third term is the third term (a + 2d) of the arithmetic progression, and the seventh term is the seventh term (a + 6d) of the arithmetic progression. We can set up another equation based on this:
a + 2d = a + ar^2
a + 6d = a + 2ar^2
Now, we can solve the equations simultaneously to find the values of "a", "r", and "d".
From the first equation:
a + ar + ar^2 = 28
=> a(1 + r + r^2) = 28
=> a = 28/(1 + r + r^2)
From the second equation:
a + 2d = a + ar^2
=> 2d = ar^2
=> d = a * r^2 / 2
From the third equation:
a + 6d = a + 2ar^2
=> 6d = 2ar^2
=> d = a * r^2 / 3
Now, equating the expressions for "d":
a * r^2 / 2 = a * r^2 / 3
2/3 = r^2
r^2 = 2/3
r = √(2/3)
Substituting the value of "r" in the equation for "a":
a = 28/(1 + r + r^2)
a = 28/(1 + √(2/3) + 2/3)
a = 28/(1 + √2/√3 + 2/3)
a = 84/(3 + √2√3 + 2)
a = 84/(5 + √6)
Now, we can substitute "a" and "r" in the equation to find the consecutive terms of the geometric progression:
First term: a = 84/(5 + √6)
Second term: ar = 84/(5 + √6) * √(2/3)
Third term: ar^2 = 84/(5 + √6) * (√(2/3))^2
Simplifying the expressions will give us the actual values of the three numbers.