The pth term of an A.P is 20 and qth term is 10 Find the common difference
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If the m
th
term of an
m
th
term is 20
a+(m-1)d=20 --(1)
n
th
term is 10
a+(n-1)d=10-----(2)
sub (2) & (1)
a+(m-1) d - a (n-1) d = 20-10
=( m - 1 - n+1) d=10
= (m-n) d = 10
∴d=
m−n
10
substituting
a+(m−1)
m−n
10
=20
=a=20−
m−n
(m−1)10
=
m−n
20m−20n−10m+10
=
m−n
10m+10−20n
sum of (m+n) terms
2
(m+n)
[2a+(m+n−1)d]
=
2
(m+n)
[
m+n
2(10m+10−20n)
+(m+n−1)
m−n
10
]
=
2
m+n
[
m+n
20m+20−40n
+
m−n
10m+10n−1
]
=(
2
m+n
)[
(m+n)−(m−n)
20m
2
+20m−40mn−20mn−20n+40n
2
+10m
2
+10mn−m+10mn+10n
2
−n
]
= (
2
m+n
)[30+
m−n
10
]
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