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Sum of First n Terms of an AP

The sum of the first \boldsymbol{n} terms of an AP is a widely used formula in mathematics to find the total value of terms in a sequence without manually adding each term. This formula is applicable in solving problems across finance, physics, and engineering.

Formula for the Sum of First n Terms

The sum of the first \boldsymbol{n} terms of an Arithmetic Progression (AP) is denoted by \boldsymbol{S_n}. There are two primary formulas:

  1. If the first term (\boldsymbol{a}) and the common difference (\boldsymbol{d}) are known:

\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}

  1. If the first term (\boldsymbol{a}) and the last term (\boldsymbol{l}) are known:

\boldsymbol{S_n = \frac{n}{2} (a + l)}

Here:

  • \boldsymbol{S_n} = the sum of the first \boldsymbol{n} terms,
  • \boldsymbol{a} = the first term of the AP,
  • \boldsymbol{d} = the common difference between consecutive terms,
  • \boldsymbol{n} = the total number of terms,
  • \boldsymbol{l} = the last term of the AP.

Derivation of the Formula

Using the Sequence Structure

Consider an AP: \boldsymbol{a, , a + d, , a + 2d, \dots, a + (n - 1)d}.
The sum of the first \boldsymbol{n} terms can be expressed as:
\boldsymbol{S_n = a + (a + d) + (a + 2d) + \dots + (a + (n - 1)d)}.

Writing this sum in reverse order:
\boldsymbol{S_n = (a + (n - 1)d) + (a + (n - 2)d) + \dots + a}.

Add these two equations:
\boldsymbol{2S_n = n \big[2a + (n - 1)d\big]}.

Divide both sides by \boldsymbol{2}:
\displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}.

Examples with Step-by-Step Solutions

Example 1: Find the sum of the first 15 terms of the AP \boldsymbol{3, 7, 11, 15, \dots}.

Solution:

  1. Identify the first term and the common difference:
    \boldsymbol{a = 3, , d = 7 - 3 = 4}.
  2. Use the formula for the sum of the first \boldsymbol{n} terms:
    \displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}.
  3. Substitute the values of \boldsymbol{a}, \boldsymbol{d}, and \boldsymbol{n = 15}:
    \displaystyle\boldsymbol{S_{15} = \frac{15}{2} \big[2(3) + (15 - 1)(4)\big]}.
  4. Simplify the expression:
    \displaystyle\boldsymbol{S_{15} = \frac{15}{2} \big[6 + 56\big]}
    \displaystyle\boldsymbol{S_{15} = \frac{15}{2} \cdot 62 = 465}.

Answer: The sum of the first 15 terms is \boldsymbol{S_{15} = 465}.

Example 2: The sum of the first 10 terms of an AP is \boldsymbol{220}, and the first term is \boldsymbol{12}. Find the common difference.

Solution:

  1. Use the formula for the sum of the first \boldsymbol{n} terms:
    \displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}.
  2. Substitute the known values:
    \displaystyle\boldsymbol{220 = \frac{10}{2} \big[2(12) + (10 - 1)d\big]}.
  3. Simplify the equation:
    \boldsymbol{220 = 5 \big[24 + 9d\big]}
    \boldsymbol{220 = 120 + 45d}.
  4. Solve for \boldsymbol{d}:
    \boldsymbol{220 - 120 = 45d}
    \boldsymbol{100 = 45d}
    \displaystyle\boldsymbol{d = \frac{100}{45}}
    \displaystyle\boldsymbol{d = \frac{20}{9}}.

Answer: The common difference is \displaystyle\boldsymbol{d = \frac{20}{9}}.

Example 3: Find the sum of all multiples of \boldsymbol{3} between \boldsymbol{1} and \boldsymbol{100}.

Solution:

  1. Form the AP:
    The multiples of \boldsymbol{3} between \boldsymbol{1} and \boldsymbol{100} are:
    \boldsymbol{3, 6, 9, \dots, 99}.
    Here, \boldsymbol{a = 3}, \boldsymbol{d = 3}, and \boldsymbol{l = 99}.
  2. Find the number of terms (\boldsymbol{n}):
    Use the formula for the nth term:
    \boldsymbol{a_n = a + (n - 1)d}.
    Substituting \boldsymbol{a_n = 99}, \boldsymbol{a = 3}, and \boldsymbol{d = 3}:
    \boldsymbol{99 = 3 + (n - 1)(3)}
    \boldsymbol{96 = 3(n - 1)}
    \displaystyle\boldsymbol{n - 1 = \frac{96}{3}}
    \boldsymbol{n = 33}.
  3. Use the formula for the sum of the first \boldsymbol{n} terms:
    \displaystyle\boldsymbol{S_n = \frac{n}{2} (a + l)}.
    Substituting \boldsymbol{n = 33}, \boldsymbol{a = 3}, and \boldsymbol{l = 99}:
    \displaystyle\boldsymbol{S_{33} = \frac{33}{2} (3 + 99)}
    \displaystyle\boldsymbol{S_{33} = \frac{33}{2} \cdot 102}
    \displaystyle\boldsymbol{S_{33} = 1683}.

Answer: The sum of all multiples of \boldsymbol{3} between \boldsymbol{1} and \boldsymbol{100} is \boldsymbol{S_{33} = 1683}.

Example 4: Find the sum of the first 22 terms of the AP: \boldsymbol{8, 3, -2, \dots}.

Solution:

  1. Identify the values:
    \boldsymbol{a = 8}, \boldsymbol{d = 3 - 8 = -5}, \boldsymbol{n = 22}.
  2. Use the formula for the sum of \boldsymbol{n} terms:
    \displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}
  3. Substitute the values:
    \displaystyle\boldsymbol{S = \frac{22}{2} \big[2(8) + (22 - 1)(-5)\big]}
  4. Simplify the expression:
    \boldsymbol{S = 11 \big[16 + 21(-5)\big]}
    \boldsymbol{S = 11 \big[16 - 105\big]}
    \boldsymbol{S = 11(-89)}
    \boldsymbol{S = -979}

Answer: \boldsymbol{S = -979}

Example 5: If the sum of the first 14 terms of an AP is \boldsymbol{1050}, and its first term is \boldsymbol{10}, find the 20th term.

Solution:

  1. Identify the values:
    \boldsymbol{S_{14} = 1050}, \boldsymbol{n = 14}, \boldsymbol{a = 10}.
  2. Use the formula for the sum of \boldsymbol{n} terms:
    \displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}
  3. Substitute the values for the 14th term sum:
    \displaystyle\boldsymbol{1050 = \frac{14}{2} \big[2(10) + (14 - 1)d\big]}
  4. Simplify the equation:
    \boldsymbol{1050 = 7 \big[20 + 13d\big]}
    \boldsymbol{1050 = 140 + 91d}
    \boldsymbol{910 = 91d}
    \boldsymbol{d = 10}
  5. Find the 20th term using the nth term formula:
    \boldsymbol{a_n = a + (n - 1)d}
    \boldsymbol{a_{20} = 10 + (20 - 1)(10)}
    \boldsymbol{a_{20} = 10 + 190}
    \boldsymbol{a_{20} = 200}

Answer: \boldsymbol{a_{20} = 200}

Example 6: Find the sum of the first 24 terms of the list of numbers whose nth term is given by \boldsymbol{a_n = 3 + 2n}.

Solution:

  1. Expand the sequence:
    \boldsymbol{a_1 = 3 + 2(1) = 5}, \boldsymbol{a_2 = 3 + 2(2) = 7}, \boldsymbol{a_3 = 3 + 2(3) = 9}, \dots
  2. Identify the values:
    \boldsymbol{a = 5}, \boldsymbol{d = 7 - 5 = 2}, \boldsymbol{n = 24}.
  3. Use the formula for the sum of \boldsymbol{n} terms:
    \displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}
  4. Substitute the values:
    \displaystyle\boldsymbol{S_{24} = \frac{24}{2} \big[2(5) + (24 - 1)(2)\big]}
  5. Simplify the expression:
    \boldsymbol{S_{24} = 12 \big[10 + 46\big]}
    \boldsymbol{S_{24} = 12 \cdot 56}
    \boldsymbol{S_{24} = 672}

Answer: \boldsymbol{S_{24} = 672}

Example 7: A manufacturer produces 600 TV sets in the third year and 700 in the seventh year. Assuming production increases uniformly, find:

  1. The production in the first year,
  2. The production in the 10th year,
  3. The total production in the first 7 years.

Solution:

  1. Since production increases uniformly, it forms an AP. Let the first year production be \boldsymbol{a} and the common difference be \boldsymbol{d}.
    From the data:
    \boldsymbol{a + 2d = 600}
    \boldsymbol{a + 6d = 700}
    Subtract the first equation from the second:
    \boldsymbol{4d = 100 \quad \Rightarrow \quad d = 25}
    Substitute \boldsymbol{d = 25} into \boldsymbol{a + 2d = 600}:
    \boldsymbol{a + 50 = 600 \quad \Rightarrow \quad a = 550}
    Answer: Production in the first year: \boldsymbol{a = 550}.
  2. Find the production in the 10th year:
    \boldsymbol{a_{10} = a + (10 - 1)d}
    \boldsymbol{a_{10} = 550 + 9(25)}
    \boldsymbol{a_{10} = 550 + 225}
    \boldsymbol{a_{10} = 775}
    Answer: Production in the 10th year: \boldsymbol{a_{10} = 775}.
  3. Find the total production in the first 7 years:
    \displaystyle\boldsymbol{S_7 = \frac{7}{2} \big[2a + (7 - 1)d\big]}
    \displaystyle\boldsymbol{S_7 = \frac{7}{2} \big[2(550) + 6(25)\big]}
    \displaystyle\boldsymbol{S_7 = \frac{7}{2} \big[1100 + 150\big]}
    \displaystyle\boldsymbol{S_7 = \frac{7}{2} \cdot 1250}
    \boldsymbol{S_7 = 4375}

Answer: Total production in the first 7 years: \boldsymbol{S_7 = 4375}

Practice Problems

  1. Find the sum of the first \boldsymbol{20} terms of the AP \boldsymbol{4, 9, 14, 19, \dots}.
    Answer: \boldsymbol{S_{20} = 1240}.
  2. The 8th term of an AP is \boldsymbol{37}, and the sum of the first \boldsymbol{8} terms is \boldsymbol{200}. Find the first term.
    Answer: \boldsymbol{a = 13}.
  3. Find the sum of all two-digit numbers divisible by \boldsymbol{7}.
    Answer: \boldsymbol{S = 735}.

FAQs

How is the sum of an AP derived?2024-12-18T14:05:59+05:30

The sum is derived by pairing terms of the sequence in reverse order, resulting in a simplified formula.

What is the importance of the sum of an AP?2024-12-18T14:05:40+05:30

It helps in solving problems involving large sequences without manually adding all terms.

What is the formula for the sum of the first n terms of an AP?2024-12-18T14:04:26+05:30

\displaystyle\boldsymbol{S_n = \frac{n}{2} \big[2a + (n - 1)d\big]}.

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