Respuesta :
Sigma notation of sequence −10, −13, −16, … is [tex]\sum_{n=1}^{6}-(3 n+7)[/tex]
Solution:
Need to determine the sigma notation for the following sequence
−10, −13, −16, …
Let's try to build a generic formula for given sequence
The given sequence is in Arithmetic progression where first term = -10 and common difference = -3
The formula for arithmetic progression is given as:
[tex]a_n = a_1 + (n - 1)d[/tex]
Where,
[tex]a_n[/tex] is the nth term in the sequence
[tex]a_1[/tex] is the first term in the sequence
d is the common difference between the terms
Here in this sequence [tex]a_1[/tex] = -10 and d = -3
[tex]a_n = -10 + (n - 1)(-3)\\\\a_n = -10 - 3n + 3\\\\a_n = -3n - 7[/tex]
So generic formula for a term is – ( 3n + 7 )
[tex]\begin{array}{l}{\text { For } \mathrm{n}=1, \text { term is }-(3 \times 1+7)=-10} \\\\ {\text { For } \mathrm{n}=2, \text { term is }-(3 \times 2+7)=-13} \\\\ {\text { For } \mathrm{n}=3, \text { term is }-(3 \times 3+7)=-16}\end{array}[/tex]
And so on
Using sigma notation for arithmetic sequence:
[tex]\sum_{k=1}^{n} a_{k}[/tex]
So for first six terms value of n will vary from 1 to 6 and in sigma notation it can be represented as
[tex]\sum_{n=1}^{6}-(3 n+7)[/tex]
