Question

The first negative term of the A.P. 74 , 67 , 60 .............is

Answer 1

$\large\bf\underline{Given:-}$

• AP :- 74 , 67 , 60 ............

$\large\bf\underline {To \: find:-}$

• The first negative term.

$\huge\bf\underline{Solution:-}$

• AP = 74 , 67 , 60 ..

First term (a) = 74

common difference (d) = 67-74 = -7

Let the nth term of AP be the first negative term.

Then,

$\rightarrow{ \bf \: T_n < 0}$

we know that,

$\blacktriangleright \: \large {\boxed{\bf \: a_n = a+(n-1)d}}$

$\rightarrowtail \rm \: \{ 74 + (n - 1) \times ( - 7)\} < 0 \\ \\ \rightarrowtail \rm \: \{ 74 - 7n + 7\} < 0 \\ \\ \rightarrowtail \rm \:\{ 81 - 7n \} < 0\\ \\ \rightarrowtail \rm \: 81 < 7n \\ \\ \rightarrowtail \rm \:7n > 81 \\ \\ \rightarrowtail \rm \:n > \frac{81}{7} \\ \\ \rightarrowtail \rm \:n > 11.57 \\ \\ \: \: \: \: \: \bf \therefore \: n = 12$

Hence,

The 12th term is the first negative term of the given AP.

Answer 2

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{The \ 12^{th} \ term \ of \ AP \ is \ negative.}$

$\sf\orange{Given:}$

$\sf{The \ given \ A.P. \ is}$

$\sf{\implies{74, \ 67, \ 60,…}}$

$\sf\pink{To \ find:}$

$\sf{First \ negative \ term \ of \ the \ A.P.}$

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

$\sf{The \ given \ A.P. \ is}$

$\sf{\implies{74, \ 67, \ 60,…}}$

$\sf{Here, \ a=74 \ and \ d=67-74=-7}$

$\sf{The \ multiple \ of \ 7 \ which \ is \ 74 \ or \ greater \ than \ 74}$

$\sf{is \ 77.}$

$\sf{\therefore{n=\frac{77}{7}}}$

$\sf{\therefore{n=11}}$

$\sf{Concept:}$

$\sf{First \ term \ is \ 74 \ and \ 11 \ terms \ more.}$

$\sf{\therefore{n=11+1}}$

$\sf{\therefore{n=12}}$

$\sf\purple{\tt{\therefore{The \ 12^{th} \ term \ of \ AP \ is \ negative.}}}$

___________________________________

$\sf\blue{Verification:}$

$\boxed{\sf{tn=a+(n-1)d}}$

$\sf{\therefore{t12=74+(12-1)(-7)}}$

$\sf{\therefore{t12=74+11(-7)}}$

$\sf{\therefore{t12=74-77}}$

$\sf{\therefore{t12=-3}}$

$\sf{Hence, \ first \ negative \ term \ is \ 12^{th} \ term.}$