यदि a, b, c समांतर श्रेढी में हों तो सिद्ध कीजिए कि a+k,b+k,c+k समांतर श्रेढी में होंगे।
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since
a,b and c are in A.P then
2b=a+c,....... eq(1),
now
a+k, b+k and c+k will be in A.P,
if
2(b+k)=a+k + c+k,
taking
R.H.S
a+k + c+k,
a+c+2k,
from eq(1) put the value of (a+c), we get
2b+2k,
2(b+k),
hence
L.H.S=R.H.S
a,b and c are in A.P then
2b=a+c,....... eq(1),
now
a+k, b+k and c+k will be in A.P,
if
2(b+k)=a+k + c+k,
taking
R.H.S
a+k + c+k,
a+c+2k,
from eq(1) put the value of (a+c), we get
2b+2k,
2(b+k),
hence
L.H.S=R.H.S
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