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Question
If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.
Solution
Given:
\[10 a_{10} = 15 a_{15} \]
\[ \Rightarrow 10\left[ a + \left( 10 - 1 \right)d \right] = 15\left[ a + \left( 15 - 1 \right)d \right]\]
\[ \Rightarrow 10(a + 9d) = 15(a + 14d)\]
\[ \Rightarrow 10a + 90d = 15a + 210d\]
\[ \Rightarrow 0 = 5a + 120d\]
\[ \Rightarrow 0 = a + 24d\]
\[ \Rightarrow a = - 24d . . . (i)\]
To show:
\[a_{25} = 0\]
\[ \Rightarrow \text { LHS }: a_{25} = a + \left( 25 - 1 \right)d \]
\[ = a + 24d\]
\[ = - 24d + 24d \left( \text { From }(i) \right)\]
\[ = 0 = \text { RHS }\]
\[\text { Hence, proved } .\]
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