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:

$$\begin{gathered} b=a+D \\ c=b+D \\ s=15 \\ s=a+b+c=a+(a+D)+(a+2D) \\ s=3a+3D \\ {g}_1=1+a \\ {g}_2=4+b \\ {g}_3=19+c \\ {g}_2=q\cdot g_1 \\ {g}_3=q\cdot g_2 \\ 15=3a+3D \\ 5=a+D \\ {g}_2/g_1=g_3/g_2 \\ (4+b{)}^2=(1+a)\cdot (19+c) \\ (4+a+D{)}^2=(1+a)\cdot (19+a+2D) \\ (4+a+5-a{)}^2=(1+a)\cdot (19+a+2(5-a)) \\ {9}^2=(1+a)\cdot (29-a) \\ {9}^2=(1+a)\cdot (29-a) \\ {a}^2-28a+52=0 \\ p=1;q=-28;r=52 \\ D=q^2-4pr=28^2-4\cdot 1\cdot 52=576 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{28\pm \sqrt{576}}{2} \\ {a}_{1,2}=\frac{28\pm 24}{2} \\ {a}_{1,2}=14\pm 12 \\ {a}_1=26 \\ {a}_2=2 \\ a=a_2=2 \\ D=5-a=5-2=3 \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=a+D=2+3=5$

$$\begin{gathered} c=b+D=5+3=8 \\ \text{ Verifying Solution: } \\ S=a+b+c=2+5+8=15 \\ {g}_1=1+a=1+2=3 \\ {g}_2=4+b=4+5=9 \\ {g}_3=19+c=19+8=27 \\ {Q}_1=g_2/g_1=9/3=3 \\ {Q}_2=g_3/g_2=27/9=3 \end{gathered}$$

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