Question

Q. 999) If the 3rd and the 9th terms of an A.P. are 4 and − 8 respectively, then which term of this A.P is zero.​

Answer 1

${\underline{\boxed{\red{\sf{Given \: that :}}}}}$

$\longrightarrow$${\tt{3rd \:term,\: a_3 \:= \:4}}$

$\longrightarrow$${\tt{and\: 9th\: term,\: a_9\: =\: -8}}$

${\underline{\boxed{\green{\sf{Solution :}}}}}$

We know that, the nth term of AP is;

$\longrightarrow$${\tt{a_n = a + (n -1) d}}$

Therefore,

$\longrightarrow$${\tt{a_3 = a + (3 - 1) d}}$

$\longrightarrow$${\red{\tt{4 = a + 2d ………… (i)}}}$

$\longrightarrow$${\tt{a_9 = a + (9 -1) d}}$

$\longrightarrow$${\tt{\green{-8 = a + 8d …………… (ii)}}}$

On subtracting equation (i) from (ii), we will get here,

$\longrightarrow$ ${\tt{-12 = 6d}}$

$\longrightarrow$${\boxed{\red{\tt{d = -2}}}}$

From equation (i), we can write,

$\longrightarrow$4 = a + 2 (−2)

$\longrightarrow$4 = a − 4

$\longrightarrow$ ${\boxed{\red{\tt{a = 8}}}}$

Let nth term of this A.P. be zero.

$\longrightarrow$ ${\boxed{\purple{\tt{a_n = a + (n -1) d}}}}$

$\longrightarrow$0 = 8 + (n − 1) (−2)

$\longrightarrow$0 = 8 − 2n + 2

$\longrightarrow$2n = 10

$\longrightarrow$ ${\boxed{\blue{\tt{n = 5}}}}$

${\large{\underline{\red{\tt{Hence, \:5th\: term \:of \:this\: A.P. \:is \:0 .}}}}}$