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Find the common difference of the arithmetic sequence with a4=10 and a11=45.

Sagot :

barramedasm

Answer:

The common difference of the arithmetic sequence with a₄=10 and a₁₁=45 is 5.

Step-by-step explanation:

For this problem we need two major concepts to work with. These concepts are systems of linear equations and arithmetic sequence.

The use of arithmetic sequence concept is quite obvious because the question statement specifically stated to find the common difference of an arithmetic sequence.

Systems of linear equation will be used as we analyze and form useful working equations.

The equation for solving the nth term of an arithmetic sequence is

aₙ = a₁ + (n - 1)(d)

where aₙ is the nth term, a₁ is the first term, and d is the common difference.

Given:

  • a₄ = 10
  • a₁₁ = 45

Forming useful equations:

  • aₙ = a₁ + (n - 1)(d)
  • a₄ = a₁ + (4 - 1) (d)  
  • 10 = a₁ + 3d         →   equation 1
  • a₁₁ = a₁ + (11 - 1) (d)  
  • 45 = a₁ + 10d       →   equation 2

From the main equation on the first line, we obtained the equations 1 and 2. Notice that they both have missing variables, a₁ and d. Since we now have two equations and two unknowns, we can solve this as systems of linear equation.

Solution:

a₁ =  10 - 3d   →   equation 3 (came from the rearrangement of equation 1)

Substituting equation 3 to equation 2, we have

  • 45 = (10 - 3d) + 10d
  • 45 = 10 - 3d + 10d
  • 45 - 10 = -3d + 10d
  • 35 = 7d
  • d = [tex]\frac{35}{7}[/tex]
  • d = 5

Therefore, the common difference of the arithmetic sequence with a₄=10 and a₁₁=45 is 5.

Learn more about arithmetic sequence here: https://brainly.ph/question/351795

Know more about systems of linear equations here: https://brainly.ph/question/1957623

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