The sum 47

The sum of 3 numbers in an arithmetic progression (AP) is 15. If 1,4,19 are to be added to the above numbers respectively, it formed a geometric progression (GP). Find the numbers.

Correct answer:

a = 2
b = 5
c = 8

Step-by-step explanation:

b = a + D c = b + D s = 15 s = a + b + c = a + (a + D) + (a + 2 D) s = 3 a + 3 D g_{1} = 1 + a g_{2} = 4 + b g_{3} = 19 + c g_{2} = q \cdot g_{1} g_{3} = q \cdot g_{2} 15 = 3 a + 3 D 5 = a + D g_{2} / g_{1} = g_{3} / g_{2} (4 + b)^{2} = (1 + a) \cdot (19 + c) (4 + a + D)^{2} = (1 + a) \cdot (19 + a + 2 D) (4 + a + 5 - a)^{2} = (1 + a) \cdot (19 + a + 2 (5 - a)) 9^{2} = (1 + a) \cdot (29 - a) 9^{2} = (1 + a) \cdot (29 - a) a^{2} - 28 a + 52 = 0 p = 1; q = - 28; r = 52 D = q^{2} - 4 p r = 2 8^{2} - 4 \cdot 1 \cdot 52 = 576 D > 0 a_{1, 2} = \frac{- q \pm D}{2 p} = \frac{2 8 \pm 5 7 6}{2} a_{1, 2} = \frac{2 8 \pm 2 4}{2} a_{1, 2} = 14 \pm 12 a_{1} = 26 a_{2} = 2 a = a_{2} = 2 D = 5 - a = 5 - 2 = 3

Our quadratic equation calculator calculates it.

b = a + D = 2 + 3 = 5

c = b + D = 5 + 3 = 8 Verifying Solution: S = a + b + c = 2 + 5 + 8 = 15 g_{1} = 1 + a = 1 + 2 = 3 g_{2} = 4 + b = 4 + 5 = 9 g_{3} = 19 + c = 19 + 8 = 27 Q_{1} = g_{2} / g_{1} = 9 / 3 = 3 Q_{2} = g_{3} / g_{2} = 27 / 9 = 3

Did you find an error or inaccuracy? Feel free to write us. Thank you!

Tips for related online calculators

Are you looking for help with calculating roots of a quadratic equation?
Do you have a linear equation or system of equations and are looking for its solution? Or do you have a quadratic equation?