Question

the sum of the4thand8th terman apis24 andthe sumof the6thand10th termis44.find the first three term of the. ap​

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

$\bigstar{\bold{First\:three\:terms=-13,-8,-3}}$

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• Sum of fourth and eighth term is 24
• Sum of sixth and tenth term is 44

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• First three terms of the A.P

$\Large{\underline{\underline{\bf{Solution:}}}}$

→  The fourth term of an A.P is given by

    a₄ = a₁ + (4 - 1)d

    a₄ = a₁ + 3d---(1)

→ The eighth term of an A.P is given by

   a₈ = a₁ + (8-1)d

   a₈ = a₁ + 7d----(2)

→ By given we know that,

  a₄ + a₈ = 24

→ Substitute the values from equation 1 and 2,

  a₁ + 3d + a₁ + 7d = 24

  2a₁ + 10d = 24

→ Divide the equation by 2

  a₁ + 5d = 12--------(3)

→ Now in the second case,

→ The 6th term of an A.P is given by,

   a₆ = a₁ + (6 - 1)d

   a₆ = a₁ + 5d-----(4)

→ The 10th term of an A.P is given by

  a₁₀ = a₁ + (10 - 1 )d

  a₁₀ = a₁ + 9d----(5)

→ By given we know that,

  a₆ + a₁₀ = 44

  a₁ + 5d + a₁ + 9d = 44

  2a₁ + 14d = 44

→ Divide the whole equation by 2,

  a₁ + 7d = 22------(6)

→ Solve equation 3 and 6 by elimination method

  a₁ + 7d = 22

  a₁ + 5d = 12

         2d = 10

            d = 5

→ Hence common difference of the A.P is 5.

→ Substitute the value of d in equation 6

   a₁ + 7 × 5 = 22

   a₁ = 22 - 35

   a₁ = -13

→ Hence the first term of the A.P is -13.

→ Second term of the A.P (a₂ ) = a₁ + d

  Second term = -13 + 5

  Second term = -8

→ Third term of the A.P is given by

   Third term (a₃) = a₂ + d

   Third term = -8 + 5 = -3

→ Hence the first three terms of the A.P is -13, -8, -3

$\boxed{\bold{First\:three\:terms=-13,-8,-3}}$

$\Large{\underline{\underline{\bf{Verification:}}}}$

→ a₄  +a₈ = 24

  a₃ + d + a₁ + 7d = 24

  -3 + 5 + -13 + 7 × 5 = 24

  -11 + 35 = 24

   24 = 24

→ a₆ + a₁₀ = 44

  a₁ + 5d + a₁ + 9d = 44

 -13 + 5 ×5 + -13 + 9 × 5 = 44

 -13 + 25 + -13 + 45 = 44

  12 + 32 = 44

   44 = 44

→ Hence verified.