Question 1: The sum of three terms of an A.P. is $\displaystyle 21$ and the product of the first and the third terms exceeds the second term by $\displaystyle 6$, find three terms.

$\displaystyle \text{Let the three terms of the A.P. be } a-d, a, a+d$

$\displaystyle \therefore (a-d) + a + ( a+d) = 21$

$\displaystyle \Rightarrow 3a = 21$

$\displaystyle \Rightarrow a = 7$

$\displaystyle \text{Also } (a-d)(a+d) - a = 6$

$\displaystyle \Rightarrow a^2 - d^2 - a = 6$

$\displaystyle \Rightarrow 49 - d^2-7 = 6$

$\displaystyle \Rightarrow d^2 = 36$

$\displaystyle \Rightarrow d = \pm 6$

$\displaystyle \text{When } a = 7, d = 6 , \text{the three terms are } 1, 7, 13$

$\displaystyle \text{When } a = 7, d = -6 , \text{the three terms are } 13, 7, 1$

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Question 2: Three numbers are in A.P. If the sum of these numbers be $\displaystyle 27$ and the product $\displaystyle 548$, find the numbers.

$\displaystyle \text{Let the three terms of the A.P. be } a-d, a, a+d$

$\displaystyle \therefore (a-d) + a + ( a+d) = 27$

$\displaystyle \Rightarrow 3a = 27$

$\displaystyle \Rightarrow a = 9$

$\displaystyle \text{Also } (a-d) \cdot a \cdot (a+d) = 648$

$\displaystyle \Rightarrow a(a^2 - d^2) = 648$

$\displaystyle \Rightarrow 9(81-d^2) = 648$

$\displaystyle \Rightarrow (81-d^2) = 72$

$\displaystyle \Rightarrow d^2 = 9$

$\displaystyle \Rightarrow d = \pm 3$

$\displaystyle \text{When } a = 9, d = 3 , \text{the three terms are } 6, 9, 12$

$\displaystyle \text{When } a = 9, d = -3 , \text{the three terms are } 12, 9, 6$

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Question 3: Find the four numbers in A.P., whose sum is $\displaystyle 50$ and in which the greatest number is $\displaystyle 4$ times the least.

$\displaystyle \text{Let the four numbers be } a-3d, a-d, a+d, a+3d$

$\displaystyle \therefore (a-3d) + ( a-d) + (a+d) + ( a+3d) = 50$

$\displaystyle \Rightarrow 4a = 50$

$\displaystyle \Rightarrow a = \frac{25}{2}$

$\displaystyle \text{Also } a+3d = 4 ( a - 3d)$

$\displaystyle \Rightarrow a+3d = 4a - 12 d$

$\displaystyle \Rightarrow 15 d = 3a$

$\displaystyle \Rightarrow a = 5d$

$\displaystyle \Rightarrow d = \frac{25}{2} \times \frac{1}{5} = \frac{5}{2}$

$\displaystyle \therefore$ the terms are $\displaystyle \Big( \frac{25}{2}- 3 \times \frac{5}{2} \Big), \Big( \frac{25}{2}- \frac{5}{2} \Big) , \Big( \frac{25}{2}+ \frac{5}{2} \Big) , \Big( \frac{25}{2}+ 3 \times \frac{5}{2} \Big) \text{ or } 5, 10, 15, 20$

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Question 4: The sum of three numbers in A.P. is $\displaystyle 12$ and the sum of their cubes is $\displaystyle 288$. Find the numbers.

$\displaystyle \text{Let the three terms of the A.P. be } a-d, a, a+d$

$\displaystyle \therefore (a-d) + a + ( a+d) = 12$

$\displaystyle \Rightarrow 3a = 12$

$\displaystyle \Rightarrow a = 4$

$\displaystyle \text{Also } (a-d)^3 + a^3 + ( a+d)^3 = 288$

$\displaystyle \Rightarrow a^3 - d^3 -3a^2d + 3ad^2 + a^3 +a^3 +d^3 + 3a^2d+ 3ad^2 = 288$

$\displaystyle \Rightarrow 3a^3 + 6ad^2 = 288$

$\displaystyle \Rightarrow 3(4)^3 + 6(4) d^2 = 288$

$\displaystyle \Rightarrow 192 + 24d^2 = 288$

$\displaystyle \Rightarrow 24d^2 = 96$

$\displaystyle \Rightarrow d= \pm 2$

$\displaystyle \text{When } a=4, d = 2 , \text{the three terms are } 2,4, 6$

$\displaystyle \text{When } a = 4, d = -2 , \text{the three terms are } 6, 4, 2$

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Question 5: If the sum of three numbers in A.P. is $\displaystyle 24$ and their product $\displaystyle 440$, find the numbers.

$\displaystyle \text{Let the three terms of the A.P. be } a-d, a, a+d$

$\displaystyle \therefore (a-d) + a + ( a+d) = 24$

$\displaystyle \Rightarrow 3a = 24$

$\displaystyle \Rightarrow a = 8$

$\displaystyle \text{Also, } a(a-d)(a+d) = 440$

$\displaystyle \Rightarrow a(a^2 - d^2) = 440$

$\displaystyle \Rightarrow 8(64-d^2) = 440$

$\displaystyle \Rightarrow d^2 = 9$

$\displaystyle \Rightarrow d = \pm 3$

$\displaystyle \text{When } a = 8, d = 3 , \text{the three terms are } 5, 8, 11$

$\displaystyle \text{When } a = 8, d = -3 , \text{the three terms are } 11, 8, 5$

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Question 6: The angles of a quadrilateral are in A.P. whose common difference is $\displaystyle 10^{\circ}$. Find the angles.

$\displaystyle \text{Let the angles be } A, (A+d), (A+2d), (A+3d)$
$\displaystyle \text{Given } d = 10^{\circ}$
$\displaystyle \text{Sum of the angles } A+ (A+d)+ (A+2d)+ (A+3d) = 360^{\circ}$
$\displaystyle \Rightarrow 4A + 60^{\circ} = 360^{\circ}$
$\displaystyle \Rightarrow A = 75^{\circ}$
$\displaystyle \text{Therefore the angles are } 75^{\circ}, 85^{\circ}, 95^{\circ}, 105^{\circ}$.