If there are $11$ arithmetic means between $20$ and $80$ , identify the value of the fourth mean.

40

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45

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60

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32

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Solution

The correct option is C $40$
Let the AM to be inserted between $20$ and $80$ are
$A_{1}, A_{2}, A_{3}, . . . . A_{11}$
Then we have the series $20, A_{1}, A_{2}, A_{3}, . . . . A_{11}, 80$ forms an arithmetic progression
$a_{n} = a + (n - 1) d$ ...... ( $n^{t h}$ term of AP)
$80 = 20 + (13 - 1) d 60 = 12 d d = 5$
$4^{t h}$ mean is
$a_{5}$ $= a + 4 d = 20 + 4 \times 5 = 40$

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Class	0 – 20	20 – 40	40 – 60	60 – 80	80 – 100
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C.I.	$0 - 20$	$20 - 40$	$40 - 60$	$60 - 80$	$80 - 100$
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The n arithmetic means between 20 and 80 are such that the first mean : last mean = 1 : 3. Find the value of n.

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