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ACTIVITY CARD 1: Find the sum of the following arithmetic sequence. Show your complete solution.
1. Even integers between 1 and 101.
2, 4, 6, 8, 10, …, 100
2. Odd integers between 0 and 100.
1, 3, 5, 7, 9, …, 99
3. Multiples of 10 from 10 to 200.
10, 20, 30, 40, …, 200​

Sagot :

DIRECTIONS:

Find the sum of the following arithmetic sequence. Show your complete solution.

  1. Even integers between 1 and 101.
  2. Odd integers between 0 and 100.
  3. Multiples of 10 from 10 to 200.

ANSWERS:

  1. The sum is 2550.
  2. The sum of 2500.
  3. The sum is 2100.

SOLUTION:

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For Number 1,

» List the even integers between 1 and 101.

[tex] \boxed {\begin{array}{c} \small\underline\textsf{Even numbers from 1 to 101:} \\\\ \textsf{ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, } \\\textsf {22, 24, 26, 28, 30, 32, 34, 36, 38, 40, } \\\textsf{42, 44, 46, 48, 50, 52, 54, 56, 58, 60, } \\ \textsf{ 62, 64, 66, 68, 70, 72, 74, 76, 78, 80,} \\\textsf{ 82, 84, 86, 88, 90, 92, 94, 96, 98, and 100. } \\\end{array}} [/tex]

» The sequence formed is:

[tex] \qquad \begin{array}{|c|} \large\boxed{\textsf{ 2, 4, 6, 8, 10, ..., 100 }} \\\\ \underline\textsf{Where:} \\\\ \small\textsf{The first term} \: {\sf{a_1 \: = \: 2 } } \\ \small\textsf{The number of terms} \: {\sf{n \: = \: 50 }} \\ \small\textsf{The common difference} \: {\sf{d \: = \: 2 }} \end{array} [/tex]

» Solve for the sum of the terms.

  • [tex] \sf{S_n =\frac{n}{2} [2a_1 + (n-1)d] } [/tex]

  • [tex] \sf{S_{50} =\frac{50}{2} [2(2) + (50-1)2] } [/tex]

  • [tex] \sf{S_{50} = 25 [4 + (49)2] } [/tex]

  • [tex] \sf{S_{50} = 25 (4 + 98) } [/tex]

  • [tex] \sf{S_{50} = 25 (102) } [/tex]

  • [tex] \large \therefore {\green{\sf{S_{50} = 2550 }}} [/tex]

Thus, the sum of all even integers between 1 and 101 is 2550.

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For Number 2,

» List the odd integers between 1 and 100.

[tex] \boxed {\begin{array}{c} \small\underline\textsf{Odd numbers from 1 to 100:} \\\\ \textsf{ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, } \\\textsf {21, 23, 25, 27, 29, 31, 33, 35, 37, 39, } \\\textsf{41, 43, 45, 47, 49, 51, 53, 55, 57, 59, } \\ \textsf{ 61, 63, 65, 67, 69, 71, 73, 75, 77, 79,} \\\textsf{ 81, 83, 85, 87, 89, 91, 93, 95, 97, and 99. } \\\end{array}} [/tex]

» The sequence formed is:

[tex] \qquad \begin{array}{|c|} \large\boxed{\textsf{ 1, 3, 5, 7, 9, ..., 99 }} \\\\ \underline\textsf{Where:} \\\\ \small\textsf{The first term} \: {\sf{a_1 \: = \: 1 } } \\ \small\textsf{The number of terms} \: {\sf{n \: = \: 50 }} \\ \small\textsf{The common difference} \: {\sf{d \: = \: 2 }} \end{array} [/tex]

» Solve for the sum of the terms.

  • [tex] \sf{S_n =\frac{n}{2} [2a_1 + (n-1)d] } [/tex]

  • [tex] \sf{S_{50} =\frac{50}{2} [2(1) + (50-1)2] } [/tex]

  • [tex] \sf{S_{50} = 25 [2 + (49)2] } [/tex]

  • [tex] \sf{S_{50} = 25 (2 + 98) } [/tex]

  • [tex] \sf{S_{50} = 25 (100) } [/tex]

  • [tex] \large \therefore {\green{\sf{S_{50} = 2500 }}} [/tex]

Thus, the sum of all odd integers between 1 and 100 is 2500.

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For Number 3,

» List the multiples of 10 from 10 to 200.

[tex] \qquad \boxed {\begin{array}{c} \small\underline\textsf{Multiples of 10 from 10 to 200. } \\\\ \textsf{ 10, 20, 30, 40, 50 } \\\textsf { 60, 70, 80, 90, 100 } \\\textsf{ 110, 120, 130, 140, 150, } \\ \textsf{ 160, 170, 180, 190, and 200} \\\end{array}} [/tex]

» The sequence formed is:

[tex] \qquad \begin{array}{|c|} \large\boxed{\textsf{ 10, 20, 30, 40, 50, ..., 200 }} \\\\ \underline\textsf{Where:} \\\\ \small\textsf{The first term} \: {\sf{a_1 \: = \: 10 } } \\ \small\textsf{The number of terms} \: {\sf{n \: = \: 20 }} \\ \small\textsf{The common difference} \: {\sf{d \: = \: 10 }} \end{array} [/tex]

» Solve for the sum of the terms.

  • [tex] \sf{S_n =\frac{n}{2} [2a_1 + (n-1)d] } [/tex]

  • [tex] \sf{S_{20} =\frac{20}{2} [2(10) + (20-1)10] } [/tex]

  • [tex] \sf{S_{20} = 10 [20 + (19)10] } [/tex]

  • [tex] \sf{S_{20} = 10 (20 + 190) } [/tex]

  • [tex] \sf{S_{20} = 10 (210) } [/tex]

  • [tex] \large \therefore {\green{\sf{S_{20} = 2100 }}} [/tex]

Thus, the sum of all multiples of 10 from 10 to 200 is 2100.

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