Answer: 117
Step-by-step explanation:
Formula for arithmetic sequence is:
= a + (n - 1)d
22nd term = a + (n - 1)d
= a + (22 - 1)d = a + 21d
where, a = 12
d = 17 - 12 = 5.
Therefore, 22nd term will be:
= a + 21d
= 12 + 21(5)
= 12 + 105
= 117
The 22nd term is 117
The next two term in patterns is below
3,6,5,10,9,18,17,34,33,66
Answer:
No, because using the distributive property, you would get 5b+15 and 3b+15. The number of b you have is different.
Given:
![(ax+2)(bx+7)=15x^2+cx+14](https://tex.z-dn.net/?f=%28ax%2B2%29%28bx%2B7%29%3D15x%5E2%2Bcx%2B14)
And
![a+b=8](https://tex.z-dn.net/?f=a%2Bb%3D8)
Required:
To find the two possible values of c.
Explanation:
Consider
![\begin{gathered} (ax+2)(bx+7)=15x^2+cx+14 \\ abx^2+7ax+2bx+14=15x^2+cx+14 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%28ax%2B2%29%28bx%2B7%29%3D15x%5E2%2Bcx%2B14%20%5C%5C%20abx%5E2%2B7ax%2B2bx%2B14%3D15x%5E2%2Bcx%2B14%20%5Cend%7Bgathered%7D)
So
![\begin{gathered} ab=15-----(1) \\ 7a+2b=c \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20ab%3D15-----%281%29%20%5C%5C%207a%2B2b%3Dc%20%5Cend%7Bgathered%7D)
And also given
![a+b=8---(2)](https://tex.z-dn.net/?f=a%2Bb%3D8---%282%29)
Now from (1) and (2), we get
![\begin{gathered} a+\frac{15}{a}=8 \\ \\ a^2+15=8a \\ \\ a^2-8a+15=0 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20a%2B%5Cfrac%7B15%7D%7Ba%7D%3D8%20%5C%5C%20%20%5C%5C%20a%5E2%2B15%3D8a%20%5C%5C%20%20%5C%5C%20a%5E2-8a%2B15%3D0%20%5Cend%7Bgathered%7D)
![a=3,5](https://tex.z-dn.net/?f=a%3D3%2C5)
Now put a in (1) we get
![\begin{gathered} (3)b=15 \\ b=\frac{15}{3} \\ b=5 \\ OR \\ b=\frac{15}{5} \\ b=3 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%283%29b%3D15%20%5C%5C%20b%3D%5Cfrac%7B15%7D%7B3%7D%20%5C%5C%20b%3D5%20%5C%5C%20OR%20%5C%5C%20b%3D%5Cfrac%7B15%7D%7B5%7D%20%5C%5C%20b%3D3%20%5Cend%7Bgathered%7D)
We can interpret that either of a or b are equal to 3 or 5.
When a=3 and b=5, we have
![\begin{gathered} c=7(3)+2(5) \\ =21+10 \\ =31 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20c%3D7%283%29%2B2%285%29%20%5C%5C%20%3D21%2B10%20%5C%5C%20%3D31%20%5Cend%7Bgathered%7D)
When a=5 and b=3, we have
![\begin{gathered} c=7(5)+2(3) \\ =35+6 \\ =41 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20c%3D7%285%29%2B2%283%29%20%5C%5C%20%3D35%2B6%20%5C%5C%20%3D41%20%5Cend%7Bgathered%7D)
Final Answer:
The option D is correct.
31 and 41