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The sum of the 10 terms of arithmetic sequence is 55 and the first term is 1. Which is the common ratio?

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UK2019IN

✏️ARITHMETIC SERIES

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[tex]\underline{\mathbb{PROBLEM:}}[/tex]

  • The sum of the 10 terms of arithmetic sequence is 55 and the first term is 1. What is the common difference?

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[tex]\underline{\mathbb{ANSWER:}}[/tex]

[tex]\qquad \LARGE \rm» \:\: \green{d = 1}[/tex]

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[tex]\underline{\mathbb{SOLUTION:}}[/tex]

- Using the Arithmetic Series Formula to find the common difference.

[tex]\begin{aligned}&\bold{\color{lightblue}Formula:}\\&\boxed{S_n = \frac{n}{2}\big[2a_1 + (n - 1)d\big] } \end{aligned}[/tex]

  • [tex]\begin{aligned}{S_{10} = \frac{10}{2}\big[2(1) + (10- 1)d\big] } \end{aligned}[/tex]

  • [tex]\begin{aligned}{55 = \frac{10}{2}\big[2(1) + (10- 1)d\big] } \end{aligned}[/tex]

  • [tex]\begin{aligned}{55 = \frac{10}{2}\big[2(1) + (9)d\big] } \end{aligned}[/tex]

  • [tex]\begin{aligned}{55 = \frac{10}{2}\big[2 + 9d\big] } \end{aligned}[/tex]

  • [tex]55 = 5\big[2 + 9d\big][/tex]

  • [tex]55 = 10 + 45d[/tex]

  • [tex]55 - 10 = 45d [/tex]

  • [tex]45 = 45d [/tex]

  • [tex]\begin{aligned}{ \frac{45}{45} = \frac{ \cancel{45}d}{ \cancel{45}} } \end{aligned}[/tex]

  • [tex]1 = d[/tex]

[tex]\therefore[/tex] The common difference is 1.

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