# arithmetic

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there are n arithmetic means between 5 and 32 such that the ratio between the 3rd and 7th means is 7:13,find the value of n.

• arithmetic -

the sequence would be

5 , 5+d , 5+2d , 5+3d , .... , 5 + (n-2)d , 32
for n+2 terms

3rd mean = 4th term = a+3d
7th mean = 8th term = a + 7d
but a = 5

(5+3d)/(5+7d) = 7/13
35 + 49d = 65 +39d
10d = 30
d = 30/10 = 3

so a + ((n+2)-1)d = 32
5 + (n+1)(3) = 32
3n+3 = 27
3n = 24
n = 8

check:

5 8 11 14 17 20 23 26 29 32

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