Question

Which term of an ap - 7 - 12 - 17 - 22 will be - 82? Is -180 term of an ap give reason for your answer?

Answer 1

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

$\textbf{\underline{Arithmetic\;Progression}}$

-7,-12,-17,-22

$\textbf{\underline{First\;Term}}$

= -7

$\textbf{\underline{Common\; Difference}}$

= -12 - (-7)

= -5

$\textbf{\underline{Assume}}$

${\boxed{\sf\:{n^{th}\;term\;be\;-82}}}$

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

-82 = -7 + (n - 1)(-5)

-82 + 7 = (n - 1)(-5)

-75 = (n - 1)(-5)

$\tt{\rightarrow n-1=\dfrac{-75}{-5}}$

n - 1 = 15

n = 15 + 1

n = 16

$\textbf{\underline{Hence,}}$

${\boxed{\sf\:{-82\;is\;the\;16^{th}\;term\;of\;an\;AP}}}$

$\textbf{\underline{Assume}}$

${\boxed{\sf\:{m^{th}\;term\;be\;-180}}}$

-180 = -7 + (m - 1)(-5)

-173 = (m - 1)(-5)

$\tt{\rightarrow m-1=\dfrac{-173}{-5}}$

$\tt{\rightarrow m-1=\dfrac{173}{5}}$

$\tt{\rightarrow m=\dfrac{173}{5}+1}$

$\tt{\rightarrow m=\dfrac{178}{5}}$

${\boxed{\sf\:{nth\;term\;never\;be\;in\;fraction}}}$

$\textbf{\underline{Therefore}}$

$\Large{\boxed{\sf\:{-180\;is\;not\;the\;term\;of\;AP}}}$

Answer 2

Answer: 16 th term of the given ap is -82 and -180 will not be a term in the given ap

Step-by-step explanation:

given :  -7, -12 ,-17 ,-22........

common difference 'd ' is $a_{2} -a_{1}$

here $a_{2}$ is -12

and  $a_{1}$ is -7

so 'd ' is :

-12-(-7)

-12 +7 = -5

so therefore 'd' is -5

now ,

$a_{n} = a + (n - 1) d$

(1) $a_{n} = -82$

-82 = -7 + ( n - 1 ) -5

-82= -7 - 5n +5

-82 = -2 - 5n

-82 +2 = - 5n

-80 = -5n

n = -80/-5

n   = 16

-82 is the 16th term in the ap

(2) $a_{n} =  -180$

-180 = -7 + ( n - 1 ) -5

-180= -7 - 5n +5

-180 = -2 - 5n

-180 +2 = - 5n

-178 = -5n

n = -178/-5

n   =  32.6

-180 is not a term of given ap since " n " value cannot be negative , fractional , decimal .