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# If the $$3^{rd}$$ and the $$9^{th}$$ terms of an AP are 4 and –8 respectively, which term of this AP is zero?

Answer :

It is given that $$3^{rd}$$ and $$9^{th}$$ term of AP are 4 and –8 respectively.

It means $$a_3 = 4$$ and $$a_9 = -8$$

Using formula $$a_n = a + (n - 1)d$$, to find $$n^{th}$$ term of arithmetic progression,

$$\Rightarrow$$ 4 = a + (3 – 1)d and,
–8 = a + (9 – 1)d
$$\Rightarrow$$ - 4 = a + 2d and,
-8 = a + 8d

These are equations in two variables.

Using equation 4 = a + 2d,
we can say that a = 4 - 2d

Putting value of a in other equation -8 = a + 8d,

$$\Rightarrow$$ -8 = 4 - 2d + 8d
$$\Rightarrow$$ -12 = 6d
$$\Rightarrow$$ d = -2

Putting value of d in equation -8 = a + 8d,

$$\Rightarrow$$ -8 = a + 8 (-2)
$$\Rightarrow$$ -8 = a – 16
$$\Rightarrow$$ a = 8

Therefore, first term = a = 8 and Common Difference = d = -2

We want to know which term is equal to zero.

Using formula $$a_n = a + (n - 1)d$$ , to find $$n^{th}$$ term of arithmetic progression,

$$\Rightarrow$$ 0 = 8 + (n - 1) (-2)
$$\Rightarrow$$ 0 = 8 - 2n + 2
$$\Rightarrow$$ 0 = 10 - 2n
$$\Rightarrow$$ 2n = 10
$$\Rightarrow$$ n = 5

Therefore, $$5^{th}$$ term is equal to 0.