Question

Find the rule
5. 11
6. 13
8. n
10. 21​

Answer 1

Lets form a equation

• (5,11)
• (6,13)

Slope:-

• m=13-11/6-5=2

Equation:-

• y-11=2(x-5)

So

Put x=8

• n-11=2(8-5)
• n-11=2(3)
• n-11=6
• n=17

Answer 2

n = 17

Explanation:

Method 1

The difference between the 5th and 6th terms is 13 - 11 = 2

Therefore, the next term can be found by adding 2 each time.

If we continue with this sequence, we get:

5 = 11

6 = 13

7 = 13 + 2 = 15

8 = 15 + 2 = 17

9 = 17 + 2 = 19

10 = 19 + 2 = 21

So n = 17

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Method 2

Arithmetic sequence

General form of an arithmetic sequence:
`a_n=a+(n-1)d`

where:

• 
  `a_n` is the nth term
• a is the first term
• d is the common difference between terms

Given terms of the sequence:

`a_5=11\\a_6=13\\a_8=n\\a_(10)=21`

The common difference can be found by subtracting one term from the next term:

`d=13-11=2`

To find a, substitute the found value of d into the equation for one of the given terms:

`\begin{aligned}a_5 =a+(5-1)2 & =11\\ a+8 & =11\\ a & = 3\end{aligned}`

Therefore, the formula to find the nth term is:

`\implies a_n=3+(n-1)2`

`\implies a_n=2n+1`

So, the 8th term is:

`\implies a_8=2(8)+1=17`