SELECT * FROM question_mgmt as q WHERE id=1013 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='2' AND status=1 ORDER BY CAST(question_no AS UNSIGNED)

Question:

Between 1 and 31, m numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of 7th and (m – 1)thnumbers is 5:9. Find the value of m.

Answer:

Let A1, A2, … Am be m numbers such that 1, A1, A2, … Am, 31 is an A.P.

Here, a = 1, b = 31, n = m + 2

∴ 31 = 1 + (m + 2 – 1) (d)

⇒ 30 = (m + 1) d

rightwards double arrow d equals fraction numerator 30 over denominator m plus 1 end fraction space space space space space space space space... left parenthesis 1 right parenthesis

A1 = a + d

A2 = a + 2d

A3 = a + 3d …

∴ A7 = a + 7d

Am–1 = a + (m – 1) d

According to the given condition,

fraction numerator a plus 7 d over denominator a plus open parentheses m minus 1 close parentheses d end fraction equals 5 over 9 rightwards double arrow fraction numerator 1 plus 7 open parentheses begin display style fraction numerator 30 over denominator open parentheses m plus 1 close parentheses end fraction end style close parentheses over denominator 1 plus open parentheses m minus 1 close parentheses open parentheses begin display style fraction numerator 30 over denominator m plus 1 end fraction end style close parentheses end fraction equals 5 over 9 space space space space space space space space space space space space space space space space open square brackets F r o m space left parenthesis 1 right parenthesis close square brackets rightwards double arrow fraction numerator m plus 1 plus 7 open parentheses 30 close parentheses over denominator m plus 1 plus 30 open parentheses m minus 1 close parentheses end fraction equals 5 over 9 rightwards double arrow fraction numerator m plus 1 plus 210 over denominator m plus 1 plus 30 m minus 30 end fraction equals 5 over 9 rightwards double arrow fraction numerator m plus 211 over denominator 31 m space minus 29 end fraction equals 5 over 9 rightwards double arrow 9 m space plus space 1899 space equals space 155 m space minus 145 rightwards double arrow 155 m space minus 9 m space equals space 1899 space plus 145 rightwards double arrow 146 m equals 2044 rightwards double arrow m equals 14

Thus, the value of m is 14.


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