Question

Show work and explain with formulas.

Answer 1

14 Answer: 28  

Step-by-step explanation:

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

15A Answer: 49  

Step-by-step explanation:

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

15B Answer: 12  

Step-by-step explanation:

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

16 Answer: $\sum\limits^5_{n=1} 3n-2$

Step-by-step explanation:

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