Question

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Topic - Arthematic Progression

If the 3rd term of an AP is 4
and 9th term is -8
then which term is 0

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Answer 1

$\large \bf \clubs \:  Given :-$

For an A.P :

3rd Term =  $\sf a_3$ = 4

9th term = $\sf a_9$ = - 8

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$\large \bf \clubs \:  To \: Find :-$

Which term of the corresponding A.P is 0

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$\large \bf \clubs \:  Main \: Formula :-$

☆ nth term of an A.P is given by :

$\bf \large a_n = a + (n - 1)d$

Where :

• a = First Term

• d = Common Difference

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$\large \bf \clubs \:  Solution :-$

Let a and d be the first term and common difference of the corresponding A.P. respectively.

Hence ,

$\large \sf a_3 = 4 \\  \\  \large\bf{ a + 2d = 4} \: -  -  -  - (1)$

$\large \sf a_9 =  - 8 \\  \\  \large\bf{ a + 8d =  - 8} \: -  -  -  - (2)$

Subtracting (1) From (2) :

$:\longmapsto\sf6d =  - 12 \\  \\ \LARGE \purple{ :\longmapsto  \underline {\boxed{{\bf d =   - 2} }}}$

Putting Value of d in (1) :

$\sf a + 2 \times ( - 2) = 4 \\  \\  \LARGE\purple{ :\longmapsto\underline  {{{\boxed{{\bf a = 8} }}}}}$

Now,

Let nth term of the A.P. is zero :

Hence ,

$\sf a_n = 0 \\  \\ :\longmapsto\sf a + (n - 1)d = 0 \\  \\ :\longmapsto \sf8 + (n - 1)( - 2) = 0 \\  \\ :\longmapsto\sf8 - 2n + 2 = 0 \\  \\ :\longmapsto\sf2n = 10 \\  \\ \LARGE \purple{ :\longmapsto  \underline {\boxed{{\bf n = 5} }}}$

$\underline{\underline \pink{\pmb{ \mathfrak{ \text{H}ence \:  5th  \: term \:  of  \: the \:  \bold{ A.P} \:  i s \:  \bf 0}}}}$

$\Large\red{\mathfrak{  \text{W}hich \:\:is\:\: the\:\: required} }\\ \LARGE \red{\mathfrak{ \text{ A}nswer.}}$

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Answer 2

$\underline{\purple{\ddot{\Mathsdude}}}$

☆☆Answered by Rohith kumar maths dude :-

▪We know that ,

Ap=an=a+(n-1)d

☆☆Here given:-

Third term is 4 9th term is -8

a3=a+(3-1)d a9=a+(9-1)d

4=a+2d -8=a+8d

4-2d=a -8-8d=a

a=4-2d ( i) a= -8d-8 (ii)

☆☆☆From 1 and 2:-

4-2d=-8d-8

-2d+8d=-8-4

6d=-12

d=-2

☆☆Putting value of d in 2nd equation

we get,

a=4-2d

a=4-2 (-2)

a= 4+4

a=8

☆Here we need to check that which term of an ap is zero.

■So,

●an=0

●Also a=8 and d=-2

▪In the given formula

an=a+(n-1)d

☆Applying the values in formula

☆we get,

0=8+(n-1)×-2

0=8-2n+2

0=10-2n

2n=10

n=10/2

n=5.

▪☆Hence 5th term of an ap is zero .

☆☆Hope it helps u mate :-

☆☆Thank you.