Question

Sum of three numbers of an ap is 27 and their squares is 293 find the numbers

Answer 1

$\huge{\underline{\underline{\red{\mathfrak{SoLuTiOn :}}}}}$

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➠ Given :

Three numbers are in A. P whose sum is 27 and sum of their squares are 293.

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➠ To Find :

We have to find the three numbers.

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➠ Solution :

Let the three numbers in A. P be (a + d), (a) and (a - d)

So, A. T. Q

$\sf{(a + d) + (a) + (a - d) = 27} \\ \\ \sf{\implies a + \cancel{d} + a + a \cancel{ - d} = 27} \\ \\ \sf{\implies 3a = 27} \\ \\ \sf{\implies a = \frac{\cancel{27}}{\cancel{3}}} \\ \\ \sf{a = 9}$

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Now, their squares

$\sf{\implies (a + d)^2 + (a)^2 + (a - d)^2 = 293} \\ \\ \sf{\implies a^2 + d^2 + \cancel{2ad} + a^2 + a^2 + d^2 \cancel{ - 2ad = 293}} \\ \\ \sf{\implies 3a^2 + 2d^2 = 293} \\ \\ \bf{\: \: \: \: \: \: \: \: \: \: \: Putting \: Value \: of \: a } \\ \\ \sf{\implies 3(9)^2 + 2d^2 = 293} \\ \\ \sf{\implies 243 + 2d^2 = 293} \\ \\ \sf{\implies 2d^2 = 293 - 243} \\ \\ \sf{\implies 2d^2 = 50} \\ \\ \sf{\implies d^2 = \frac{\cancel{50}}{\cancel{2}}} \\ \\ \sf{\implies d = \sqrt{25}} \\ \\ \sf{\implies d = \pm 5}$

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When d is +5

First Number = a + d = 9 + 5 = 14

Second Number = a = 9

Third Number = a - d = 9 - 5 = 4

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When d is -5

First Number = a + d = 9 + (-5) = 4

Second Number = a = 9

Thor Number = 9 - (-5) = 14

Answer 2

The numbers are 4,9 and 14

Step-by-step explanation:

• Given that

        First number = a-d

        Second number = a

        Third number = a+d

• Given that sum of three number is 27

        Means

        First number +Second number +Third number = 27

• a-d +a +a+d =27

        3 a =27

        So a =9

• It is also given that sum of square of three number is 293

        Means

        First number² +Second number² +Third number² = 293

• (a-d)² +a² +(a+d)² =293

        a²+ d² -2ad +a² +a² + d² +2ad=293

        3a² +2d² = 293

• 3×9² +2d² = 293      (On putting a= 9)

        243 +2d² = 293

        2d² = 293 - 243

        2d² = 50

        d² = 25

        So

        d = ±5

• On taking d =+5

        First number = a-d = 9 - 5 =4

        Second number = a = 9

        Third number = a+d = 9 + 5 = 14

• On taking d =-5

        First number = a-d = 9 + 5 =14

        Second number = a = 9

        Third number = a+d = 9 - 5 = 4