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# Determine the AP whose third term is 16 and the $$7^{th}$$ term exceeds the $$5^{th}$$ term by 12.

Let first term of AP = a
Let common difference of AP = d
It is given that its 3rd term is equal to 16.

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

$$\Rightarrow$$ 16 = a + (3 - 1) (d)
$$\Rightarrow$$ 16 = a + 2d… (1)

It is also given that $$7^{th}$$ term exceeds $$5^{th}$$ term by12.

According to the given condition:

$$\Rightarrow a_7 = a_5 + 12$$
$$\Rightarrow$$ a + (7 – 1) d = a + (5 – 1) d + 12
$$\Rightarrow$$ 2d = 12
$$\Rightarrow$$ d = 6
Putting value of d in equation
$$\Rightarrow$$ 16 = a + 2d,
$$\Rightarrow$$ 16 = a + 2(6)
$$\Rightarrow$$ a = 4

Therefore, first term = a = 4

And, common difference = d = 6

Therefore, AP is 4, 10, 16, 22…