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Arithmetic Progression

Between 1 and...

Question

Between 1 and 31 m arithmetic means are inserted so that the ratio of the $7^{t h}$ and ${(m - 1)}^{t h}$ means is 5 : 9. Then the value of m is

A

12

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B

13

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C

14

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D

15

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Solution

The correct option is C
14

1, $x_{1}$ , $x_{2}$ ..... $x_{m}$ , 31 is an A.P. of (m + 2) terms.

31 = $T_{m}$ +2 = a + (m + 1)d = 1 + (m + 1)d

∴ d = $\frac{30}{(m + 1)}$

Now $\frac{x_{7}}{x_{m} - 1}$ = $\frac{5}{9}$ ∴ $\frac{T_{8}}{T_{m}}$ = $\frac{a + 7 d}{a + (m - 1) d}$ = $\frac{5}{9}$

Now put for a and d and we get m = 14.

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