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# The sum of the $$4^{th}$$ and $$8^{th}$$ terms of an AP is 24 and the sum of $$6^{th}$$ and $$10^{th}$$ terms is 44. Find the three terms of the AP.

The sum of$$4^{th}$$ and $$8^{th}$$ terms of an AP is 24 and sum of $$6^{th}$$ and $$10^{th}$$ terms is 44.

$$a_4 + a_8 = 24$$ ,and $$a_6 + a_{10} = 44$$

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

$$\Rightarrow$$ a + (4 - 1) d + [a + (8 - 1) d] = 24 and, a + (6 - 1) d + [a + (10 - 1) d] = 44
$$\Rightarrow$$ a + 3d + a + 7d = 24, and a + 5d + a + 9d = 44
$$\Rightarrow$$ 2a + 10d = 24, and 2a + 14d = 44
$$\Rightarrow$$ a + 5d = 12, and a + 7d =22

These are equations in two variables.

Using equation, a + 5d = 12, we can say that a = 12 - 5d…...(1)

Putting (1) in equation a + 7d = 22,
$$\Rightarrow$$ 12 - 5d + 7d = 22
$$\Rightarrow$$ 12 + 2d = 22
$$\Rightarrow$$ 2d = 10
$$\Rightarrow$$ d = 5

Putting value of d in equation
a = 12 - 5d,
$$\Rightarrow$$ a = 12 – 5 (5)
= 12 – 25
= -13

Therefore, first term = a = -13
and, Common difference = d = 5

Therefore, AP is –13, –8, –3, 2…

Its first three terms are –13, –8 and –3.