Question

Show work and explain with formulas.

Picture attached:

14. Evaluate

15. Find the sum:

16. Express using sigma notation: 1 + 4 + 7 + 10 + 13

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Answer 1

14 Answer: 28

Explanation:

`S_\infty=(a_1)/(1-r)\\\\\sum\limits^\infty_(n=1) 35\bigg(-(1)/(4)\bigg)^n\implies a_1=35,\ r=-(1)/(4)\\\\\\S_\infty=(35)/(1-\bigg(-(1)/(4)\bigg))=(35)/((5)/(4))=35* (4)/(5)=7* 4=\large\boxed{28}`

15A Answer: 49

Explanation:

`S_n=(a_1+a_n)/(2)\cdot n\\\\\\\sum\limits^7_(i=1) i+3\\\\a_1=1+3\quad =4\\a_7=7+3\quad =10\\n=7\\\\\\S_7=(a_1+a_7)/(2)\cdot 7\\\\\\.\ =(4+10)/(2)\cdot 7\\\\\\.\ =(14)/(2)\cdot 7\\\\\\.\ =7\cdot 7\\\\.\ =\large\boxed{49}`

15B Answer: 12

Explanation:

`S_n=(a_1+a_n)/(2)\cdot n\\\\\\\sum\limits^(10)_(k=3) k-5\\\\a_3=3-5\quad =-2\\a_(10)=10-5\quad =5\\n=8\\\\\\S_8=(a_3+a_(10))/(2)\cdot 8\\\\\\.\ =(-2+5)/(2)\cdot 8\\\\\\.\ =(3)/(2)\cdot 8\\\\\\.\ =3\cdot 4\\\\.\ =\large\boxed{12}`

16 Answer:
`\sum\limits^5_(n=1) 3n-2`

Explanation:

`\{1, 4, 7, 10, 13\}\implies a_1=1,\ d=3,\ n=5\\\\\text{The explicit rule for an arithmetic sequence is: }a_n=a_1+d(n-1)\\\\a_n=1+3(n-1)\\\\.\ =1+3n-3\\\\.\ =3n-2\\\\\large\boxed{\sum\limits^5_(n=1) 3n-2}`