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Answered

For example, how do we compute the sum of all the terms of each of the following sequences?
1. 1, 2, 3,..., 100
2. 5, 10, 15, 20,..., 50
3.-5. -2, 1, 4, ..., 31

with solution sana guyss​

Sagot :

RHOZETAIN

✏️ Arithmetic Series

[tex] {\Large{\overline{\underline{\sf{\hookrightarrow Answers:}}}}} [/tex]

  1. [tex] \sf S_{100} = 5050 [/tex]
  2. [tex] \sf S_{10} = 275 [/tex]
  3. [tex] \sf S_{13} = 169 [/tex]

Solutions

1.

Given that:

  • the first term [tex] \sf a_1 [/tex] = 1
  • the nth term [tex] \sf a_n [/tex] = 100

Solve for the common difference.

  • [tex] \sf d = a_n - a_{n-1} [/tex]
  • [tex] \sf d = a_2 - a_1 [/tex]
  • [tex] \sf d = 2 - 1 [/tex]
  • [tex] \sf d = 1 [/tex]

Find what term is 100.

  • [tex] \sf a_n = a_1 + (n - 1)d [/tex]
  • [tex] \sf 100 = 1 + (n - 1)1 [/tex]
  • [tex] \sf 100 = 1 + n - 1 [/tex]
  • [tex] \sf n = 100 [/tex]

Solve for the sum of the terms.

  • [tex] \sf{S_n =\frac{n}{2} [2a_1 + (n-1)d] } [/tex]
  • [tex] \sf{S_{100} =\frac{100}{2} [2(1) + (100-1)1] } [/tex]
  • [tex] \sf{S_{100} = 50 [2 + (99)1] } [/tex]
  • [tex] \sf{S_{100} = 50 (101) } [/tex]
  • [tex] {\sf \therefore S_{100} = {\boxed{\green{\sf{5050}}}}} [/tex]

2.

Given that:

  • the first term [tex] \sf a_1 [/tex] = 5
  • the nth term [tex] \sf a_n [/tex] = 50

Solve for the common difference.

  • [tex] \sf d = a_n - a_{n-1} [/tex]
  • [tex] \sf d = a_2 - a_1 [/tex]
  • [tex] \sf d = 10 - 5 [/tex]
  • [tex] \sf d = 5 [/tex]

Find what term is 50.

  • [tex] \sf a_n = a_1 + (n - 1)d [/tex]
  • [tex] \sf 50 = 5 + (n - 1)5 [/tex]
  • [tex] \sf 50 = 5 + 5n - 5 [/tex]
  • [tex] \sf 5n = 50 [/tex]
  • [tex] \sf n = \frac{50}{5} [/tex]
  • [tex] \sf n = 10 [/tex]

Solve for the sum of the terms.

  • [tex] \sf{S_n =\frac{n}{2} [2a_1 + (n-1)d] } [/tex]
  • [tex] \sf{S_{10} =\frac{10}{2} [2(5) + (10-1)5] } [/tex]
  • [tex] \sf{S_{10} = 5 [10 + (9)5] } [/tex]
  • [tex] \sf{S_{10} = 5 (10 + 45) } [/tex]
  • [tex] \sf{S_{10} = 5 (55) } [/tex]
  • [tex] {\sf \therefore S_{10} = {\boxed{\green{\sf{275}}}}} [/tex]

3.

Given that:

  • the first term [tex] \sf a_1 [/tex] = -5
  • the nth term [tex] \sf a_n [/tex] = 31

Solve for the common difference.

  • [tex] \sf d = a_n - a_{n-1} [/tex]
  • [tex] \sf d = a_2 - a_1 [/tex]
  • [tex] \sf d = -2 - (-5) [/tex]
  • [tex] \sf d = 3 [/tex]

Find what term is 31.

  • [tex] \sf a_n = a_1 + (n - 1)d [/tex]
  • [tex] \sf 31 = -5 + (n - 1)3 [/tex]
  • [tex] \sf 31 = -5 + 3n - 3 [/tex]
  • [tex] \sf 31 = 3n - 8 [/tex]
  • [tex] \sf 3n = 31 + 8 [/tex]
  • [tex] \sf 3n = 39 [/tex]
  • [tex] \sf n = \frac{39}{3} [/tex]
  • [tex] \sf n = 13 [/tex]

Solve for the sum of the terms.

  • [tex] \sf{S_n =\frac{n}{2} [2a_1 + (n-1)d] } [/tex]
  • [tex] \sf{S_{13} = \frac{13}{2} [2(-5) + (13-1)3] } [/tex]
  • [tex] \sf{S_{13} = \frac{13}{2} [-10 + (12)3] } [/tex]
  • [tex] \sf{S_{13} = \frac{13}{2} (-10 + 36) } [/tex]
  • [tex] \sf{S_{13} = \frac{13}{2} (26) } [/tex]
  • [tex] {\sf \therefore S_{13} = {\boxed{\green{\sf{169}}}}} [/tex]

[tex]{\: \:}[/tex]

[tex] {\huge{\overline{\sf{Hope\:It\:Helps}}}} [/tex]

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