If three numbers are in H.P., then the numbers obtrained by substracting half of the middle number from each of these are in (a) A.P. (b) G.P. (c) H.P. (d) none of these
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Step-by-step explanation:
Let the numbers be a,b,c
Given b=
a+c
2ac
.......(1)
Numbers obtained by subtracting
2
b
from each are
a−
2
b
,
2
b
,c−
2
b
Now, (a−
2
b
)(c−
2
b
)=ac−(c+a)
2
b
+
4
b
2
=ac−ac+
4
b
2
from (1)
⇒
4
b
2
=(a−
2
b
)(c−
2
b
)
∴ The new numbers are in GP
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