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if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief)
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if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd ...
Problem Statement:
The 8th term of an arithmetic progression (A.P) is equal to the geometric mean (G.M) of the 1st and 22nd term of the same A.P and the sum of the first 22 terms of the A.P is 770. Find the common difference.
Solution:
Let the first term of the A.P be 'a' and the common difference be 'd'.
Then, the 8th term of the A.P is given by:
a + 7d (as the 8th term is obtained by adding 7 times the common difference)
The 1st term of the A.P is 'a' and the 22nd term is given by:
a + 21d (as the 22nd term is obtained by adding 21 times the common difference)
Therefore, the geometric mean of the 1st and 22nd terms is:
sqrt(a(a+21d))
According to the problem statement, we have:
a + 7d = sqrt(a(a+21d))
Squaring both sides, we get:
49d^2 = a(a+21d)
Also, the sum of the first 22 terms of the A.P is given by:
S22 = (22/2)(2a + (22-1)d) = 11(2a + 21d) = 22a + 231d
According to the problem statement, we have:
S22 = 770
22a + 231d = 770
Now, we have two equations and two variables (a and d).
Solving these equations, we get:
a = 10 and d = 3
Therefore, the common difference of the A.P is 3.
The 8th term of an arithmetic progression (A.P) is equal to the geometric mean (G.M) of the 1st and 22nd term of the same A.P and the sum of the first 22 terms of the A.P is 770. Find the common difference.
Solution:
Let the first term of the A.P be 'a' and the common difference be 'd'.
Then, the 8th term of the A.P is given by:
a + 7d (as the 8th term is obtained by adding 7 times the common difference)
The 1st term of the A.P is 'a' and the 22nd term is given by:
a + 21d (as the 22nd term is obtained by adding 21 times the common difference)
Therefore, the geometric mean of the 1st and 22nd terms is:
sqrt(a(a+21d))
According to the problem statement, we have:
a + 7d = sqrt(a(a+21d))
Squaring both sides, we get:
49d^2 = a(a+21d)
Also, the sum of the first 22 terms of the A.P is given by:
S22 = (22/2)(2a + (22-1)d) = 11(2a + 21d) = 22a + 231d
According to the problem statement, we have:
S22 = 770
22a + 231d = 770
Now, we have two equations and two variables (a and d).
Solving these equations, we get:
a = 10 and d = 3
Therefore, the common difference of the A.P is 3.
Community Answer
if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd ...
D=2
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if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief)
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if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief) for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief) covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief).
if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief) for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief) covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for if 8th term of A.P is equal to G.M of 1 nd 22th term of same A.P nd sum of first 22 terms of A.P is 770 find common difference (please give solution in brief).
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