Question

Which term of the A.P. $121,117,113,.....$ is its first negative term?

Answer 1

Ans.

given A.P is

$121,117,113,.....$

first term of this A.P is $a_1=121$

second term of this A.P is $a_2=117$

common difference of an A.P is given by

$(i.e.,d=a_{n+1}-a_n)$

putting n=1 in above equation

$d=a_2-a_1=117-121=-4$

hence common difference for this A.P is $d=-4$

the nth term of an A.P is given by.

$a_n=a_1+(n-1)d$

$⟹a_n=121+(n-1)(-4)$

$⟹a_n=121-4n+4$

$⟹a_n=125-4n...eq\left(1\right)$

we have to find the term$\left(n\right)$ of this A.P such that it is the first negative term$

$⟹a_n<0$

hence from eq(1)

$⟹125-4\times n<0$

by solving this inequality

$⟹125<4\times n$

$⟹n>\frac{125}{4}$

$⟹n>31.25$ for term to be negative n must be greater than $31.25$

but we know that n (nth term) must be an integer(i.e., we can't have $31.25$th term)

hence the term corresponding to first negative number is $32$

hence $32th$ term of this A.P is its first negative term