If 7 times the 7th term of A.P is equal to 11 times the 11 term .prove that 18 term is equal to zero
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The seventh term = a7 = a + (n-1)d
= a + ( 7 - 1) d
= a + 6d
Since, 7 times the 7th term is 7(a+6d)...(1)
The eleventh term = a11 = a+(n-1)d
= a + (11-1)d
= a + 10d
Since, 11 times the 11th term is 11(a+10d) ... (2)
As it is given that, 7 times of 7th term and 11 times of 11th term are equal.
Hence,
7(a+6d) = 11(a+10d)
Multiply inside braces..
7a + 42d = 11a + 110d
7a - 11a = 110d - 42d
-4a = 68d
a = - 17d .. (3)
Therefore,
The 18th term = a18 = a+(18 - 1)d
= a+ 17d -- (4)
Hence,
Substitute a = - 17d in 4th equation..
a + 17d = - 17d + 17d
= 0
Hence proved,..
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