Question

@ Moderators  @ Stars  @ Best Users Topic - Arthematic Progression If the 3rd term of an AP is 4 and 9th term is -8 then which term is 0 ​

Answer 1

Given : An A.P 3rd Term = ${\sf{a_{3}}}$= 4 & 9th Term = ${\sf{a_{9}}}$= -8.

To Find : Find the term of the corresponding A.P is 0 ?

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Solution : Let the nth term to be x.

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$\underline{\frak{As ~we~ know ~that~:}}$

• $\boxed{\sf\purple{a_{n}~=~a~+~\bigg(n~-~1\bigg)d}}$★

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Where,

• a = First Term
• d = Common Difference

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◗Here, a and b be the first term and common difference of the corresponding A.P respectively.

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Therefore,

• ${\sf{a_{3}~=~4}}$

${\sf{a~+~2d~=~4~~~~~~~~~~~~~~~~~~~~~~~\bigg\lgroup{1~Eqⁿ}\bigg\rgroup}}$

• ${\sf{a_{9}~=~- 8}}$

${\sf{a~+~8d~=~- 8~~~~~~~~~~~~~~~~~~~~\bigg\lgroup{2~Eqⁿ}\bigg\rgroup}}$

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• Substituting (1) From (2)

$\dashrightarrow{\sf{6d~=~- 12}}$

$\dashrightarrow\boxed{\sf{d~=~- 2}}$

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• Putting Value of d in (1)

$\dashrightarrow{\sf{a~+~2~×~(- 2)~=~4}}$

$\dashrightarrow\boxed{\sf{a~=~8}}$

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Now,

• Let nth term of A.P. is zero :

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Henceforth,

$\qquad{\sf:\implies{a_{n}~=~0}}$

$\qquad{\sf:\implies{a~+~(n~-~1)d~=~0}}$

$\qquad{\sf:\implies{8~+~(n~-~1)(- 2)~=~0}}$

$\qquad{\sf:\implies{8~-~2n~+~2~=~0}}$

$\qquad{\sf:\implies{2n~=~10}}$

$\qquad{\sf:\implies{n~=~\cancel\dfrac{10}{2}}}$

$\qquad:\implies{\underline{\boxed{\frak{\pink{n~=~5}}}}}$★

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Hence,

$\therefore\underline{\bf{\underline{5th~term}}~\sf{of~the~A.P.~is~0}}$