SELECT * FROM question_mgmt as q WHERE id=1066 AND status=1 SELECT id,question_no,question,chapter FROM question_mgmt as q WHERE courseId=2 AND subId=6 AND chapterId=80 and ex_no='5' AND status=1 ORDER BY CAST(question_no AS UNSIGNED) CBSE Free NCERT Solution of 11th mathematics Sequence and Series the sum of three numbers in g p is 56 if we subt

Question:

The sum of three numbers in G.P. is 56. If we subtract 1, 7, 21 from these numbers in that order, we obtain an arithmetic progression. Find the numbers.

Answer:

Let the three numbers in G.P. be a, ar, and ar2.

From the given condition, a + ar + ar2 = 56

⇒ a (1 + r + r2) = 56

rightwards double arrow a space equals fraction numerator 56 over denominator 1 plus r plus r squared end fraction space space space space... space left parenthesis 1 right parenthesis

a – 1, ar – 7, ar2 – 21 forms an A.P.

∴(ar – 7) – (a – 1) = (ar2 – 21) – (ar – 7)

⇒ ar – a – 6 = ar2 – ar – 14

⇒ar2 – 2ar + a = 8

⇒ar2 – ar – ar + a = 8

⇒a(r2 + 1 – 2r) = 8

⇒ a (r – 1)2 = 8 … (2)

rightwards double arrow fraction numerator 56 over denominator 1 plus r plus r squared end fraction open parentheses r minus 1 close parentheses squared equals 8 space space space space space space space space space space open square brackets U sin g space left parenthesis 1 right parenthesis close square brackets

⇒7(r2 – 2r + 1) = 1 + r + r2

⇒7r2 – 14 r + 7 – 1 – r – r2 = 0

⇒ 6r2 – 15r + 6 = 0

⇒ 6r2 – 12r – 3r + 6 = 0

⇒ 6r (r – 2) – 3 (r – 2) = 0

⇒ (6r – 3) (r – 2) = 0

therefore space r space equals space 2 comma 1 half

When r = 2, a = 8

When r space equals space 1 half space comma space a space equals space 32

Therefore, when r = 2, the three numbers in G.P. are 8, 16, and 32.

When r space equals space 1 half, the three numbers in G.P. are 32, 16, and 8.

Thus, in either case, the three required numbers are 8, 16, and 32.


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