Question

The algebraic form of an arithmetic sequence 4n + 3 The sum of the first 20 terms of this sequence is

• A. 830
• B. 930
• C. 900
• D. 800

Answer 1

Answer:

C. 900

Arithmetic Progression Basics

An arithmetic progression (AP) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Algebraic Form of an AP

• The general form of an arithmetic sequence is given by a_n = a + (n-1)d, where a is the first term and d is the common difference.

• The given sequence has an algebraic form of 4n + 3. This form represents the n-th term of the sequence.

Finding the First Term and Common Difference

• To find the first term (a₁), substitute n=1 into the algebraic form: a₁ = 4(1) + 3 = 7.

• To find the second term (a₂), substitute n=2: a₂ = 4(2) + 3 = 11.

• The common difference (d) is the difference between any two consecutive terms. In this case, d = a₂ - a₁ = 11 - 7 = 4.

• Alternatively, from the form 4n + 3, the coefficient of n directly represents the common difference, so d = 4.

Sum of the First n Terms of an AP

• The formula for the sum of the first n terms of an AP is given by: S_n = n/2 * [2a + (n-1)d]

• Another form of the sum formula is: S_n = n/2 * (a₁ + a_n), where a₁ is the first term and a_n is the n-th term.

Calculating the Sum of the First 20 Terms

• We need to find the sum of the first 20 terms (S₂₀).

• Here, n = 20, a₁ = 7, and d = 4.

• Using the formula S_n = n/2 * [2a + (n-1)d]:

  • S₂₀ = 20/2 * [2(7) + (20-1)4]

  • S₂₀ = 10 * [14 + (19)4]

  • S₂₀ = 10 * [14 + 76]

  • S₂₀ = 10 * 90

  • S₂₀ = 900

• Alternatively, using the formula S_n = n/2 * (a₁ + a_n):

  • First, find the 20th term (a₂₀) using the given algebraic form a_n = 4n + 3: a₂₀ = 4(20) + 3 = 80 + 3 = 83.

  • Now, calculate the sum: S₂₀ = 20/2 * (7 + 83)

  • S₂₀ = 10 * (90)

  • S₂₀ = 900