👤
Answered

Find the sum of the first 15 terms of the sequence: - 5, - 1, 3, 7,... ​

Sagot :

KAntoineDoixgo

✏️ARITHMETIC SERIES

[tex]\red{••••••••••••••••••••••••••••••••••••••••••••••••••} [/tex]

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

  • Find the sum of the first 15 terms of the sequence: -5, -1, 3, 7, ...

[tex]\red{••••••••••••••••••••••••••••••••••••••••••••••••••} [/tex]

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

[tex] \qquad\LARGE » \tt\: \green{S_{15} = 345} [/tex]

[tex]\red{••••••••••••••••••••••••••••••••••••••••••••••••••} [/tex]

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

» Determine the common difference of the given arithmetic sequence.

[tex] \begin{align} & \bold{Formula:} \\ & \boxed{d = a_n - a_{n-1}} \end{align} [/tex]

  • [tex] d = a_2 - a_1 = \text-1 -(\text-5) = 4 [/tex]

  • [tex] d = a_3 - a_2 = 3 -(\text-1) = 4 [/tex]

  • [tex] d = a_4 - a_3 = 7 - 3 = 4 [/tex]

» Find the sum of the first 15 terms of the sequence.

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

  • [tex] S_{15} = \frac{15}{2} \big[ 2(\text-5) + 4(15 - 1) \big] \\ [/tex]

  • [tex] S_{15} = \frac{15}{2} \big[ 2(\text-5) + 4(14) \big] \\ [/tex]

  • [tex] S_{15} = \frac{15}{2} \big[ \text-10 + 56 \big] \\ [/tex]

  • [tex] S_{15} = \frac{15}{2} \big[ 46 \big] \\ [/tex]

  • [tex] S_{15} = 15 \big[ 23 \big] [/tex]

  • [tex] S_{15} = 345 [/tex]

[tex] \therefore [/tex] The sum of the first 15 terms of the given arithmetic sequence is 345.

[tex]\red{••••••••••••••••••••••••••••••••••••••••••••••••••} [/tex]

(ノ^_^)ノ

UK2019go

✏️SERIES

===============================

[tex]\large\bold{\red{PROBLEM:}}[/tex] Find the sum of the first 15 terms of the sequence: - 5, - 1, 3, 7.

[tex]\large\bold{\red{SOLUTION:}}[/tex]Find the common difference of this sequence to solve by d.

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

  • [tex] \sf {d = 7 - 3 = 4}[/tex]

  • [tex] \sf {d = 3 - (-1) = 3 + 1 = 4}[/tex]

  • [tex] \sf {d = -1 - (-5) = -1 + 5 = 4}[/tex]

- Now we have a common difference which is 4, then solve the other terms using the formula to find the sum of the first 15 terms.

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

  • [tex]\begin{gathered} \sf S_{15} = \frac{15}{2}\big[2(-5) + 4(15 - 1) \big] \end{gathered}[/tex]

  • [tex] {\begin{gathered} \sf S_{15} = \frac{15}{2}\big[-10 + 4(14) \big] \end{gathered}}[/tex]

  • [tex] {\begin{gathered} \sf S_{15} = \frac{15}{2}\big[-10 + 56 \big] \end{gathered}}[/tex]

  • [tex] {\begin{gathered} \sf S_{15} = \frac{15}{2}\big[46 \big] \end{gathered}}[/tex]

  • [tex] {\begin{gathered} \sf S_{15} = \frac{690}{2} \end{gathered}}[/tex]

  • [tex] {\begin{gathered} \sf S_{15} = 345 \end{gathered}}[/tex]

- Therefore, the first 15 terms of the arithmetic sequence is:

  • [tex] \large \boxed{\sf{\green{345}}}[/tex]

===============================

#CarryOnLearning

#LearnWithBrainly

Hindi ko po muna ma-eentertain lahat ng messages niyo sa akin. Busy po kasi ako these month and next month. God bless you po.