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The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are?
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The sum of three numbers in G.P. is 70 . If the two extemes by multipl...
Given Information:
- Let the three numbers in geometric progression be a/r, a, ar with common ratio r.
- Sum of the three numbers: a/r + a + ar = 70
- New numbers after multiplication: 4a/r, 5a, 4ar
- The products of the new numbers are in arithmetic progression.
Finding the Numbers:
Sum of the numbers:
a/r + a + ar = 70
Using the formula for the sum of a G.P., we get:
a(1/r + 1 + r) = 70
a(r^2 + 1 + r^2) = 70
2ar^2 + a = 70
a(2r^2 + 1) = 70
a = 70 / (2r^2 + 1)
Products of the new numbers:
4a/r * 4ar = 16a^2 = a(5a + a)
5a * 4ar = 20a^2 = a(4a + 5a)
The products are in A.P. if the two middle terms are twice the first term:
20a^2 = 2 * 16a^2
20a^2 = 32a^2
Solving the above equation, we get:
a = 0 (reject), a = 35
Therefore, the numbers are:
a/r = 35/r
a = 35
ar = 35r
Explanation:
- We first express the numbers in G.P. and find the sum to get the relation between the numbers.
- Then we determine the new numbers after multiplication and set up the condition for the products to be in A.P.
- Solving the equations, we find the value of 'a' which gives us the numbers in the G.P.
- Finally, we express the numbers in terms of 'a' and 'r' to get the final numbers.
- Let the three numbers in geometric progression be a/r, a, ar with common ratio r.
- Sum of the three numbers: a/r + a + ar = 70
- New numbers after multiplication: 4a/r, 5a, 4ar
- The products of the new numbers are in arithmetic progression.
Finding the Numbers:
Sum of the numbers:
a/r + a + ar = 70
Using the formula for the sum of a G.P., we get:
a(1/r + 1 + r) = 70
a(r^2 + 1 + r^2) = 70
2ar^2 + a = 70
a(2r^2 + 1) = 70
a = 70 / (2r^2 + 1)
Products of the new numbers:
4a/r * 4ar = 16a^2 = a(5a + a)
5a * 4ar = 20a^2 = a(4a + 5a)
The products are in A.P. if the two middle terms are twice the first term:
20a^2 = 2 * 16a^2
20a^2 = 32a^2
Solving the above equation, we get:
a = 0 (reject), a = 35
Therefore, the numbers are:
a/r = 35/r
a = 35
ar = 35r
Explanation:
- We first express the numbers in G.P. and find the sum to get the relation between the numbers.
- Then we determine the new numbers after multiplication and set up the condition for the products to be in A.P.
- Solving the equations, we find the value of 'a' which gives us the numbers in the G.P.
- Finally, we express the numbers in terms of 'a' and 'r' to get the final numbers.
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The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are?
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The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are?.
The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are? for CA Foundation 2024 is part of CA Foundation preparation. The Question and answers have been prepared according to the CA Foundation exam syllabus. Information about The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are? covers all topics & solutions for CA Foundation 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The sum of three numbers in G.P. is 70 . If the two extemes by multiplied each by 4 and the mean by 5,the products are in A.P. . The numbers are?.
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