Question

Show work and explain with formulas:

17. What is the sum of the series: 1 2/3 + 1 5/6 + 2 + ... + 8 1/3

18. In an arithmetic series, find the sum of the first 48 terms if the first term is -6 and the common difference is 2.

19. Find the sum of the first 8 terms of the sequence: 3, -2, -7, ...​

Answer 1

17 Answer: 205

Explanation:

`\{1(2)/(3)+1(5)/(6)+2+...+8(1)/(3)\}\implies a_1=1(2)/(3),\ d=(1)/(6)\\\\\\a_n=a_1+d(n-1)\qquad solve\ for\ n\\\\8(1)/(3)=1(2)/(3)+(1)/(6)(n-1)\\\\\\(25)/(3)=(5)/(3)+(1)/(6)n-(1)/(6)\\\\\\(50)/(6)=(10)/(6)+(1)/(6)n-(1)/(6)\\\\\\(41)/(6)=(1)/(6)n\\\\\\(41)/(6)\cdot 6=n\\\\41=n`

`\text{Now use the sum formula:}\\\\S_n=(a_1+a_n)/(2)\cdot n\\\\\\S_(41)=(1(2)/(3)+8(1)/(3))/(2)\cdot 41\\\\\\.\quad =(10)/(2)\cdot 41\\\\\\.\quad =5\cdot 41\\\\.\quad =\large\boxed{205}`

18 Answer: 1968

Explanation:

`a_1=-6,\ d=2,\ n=48, \quad \text{solve for }a_(48)\\\\a_(n)=a_1+d(n-1)\\\\a_(48)=-6+2(48-1)\\\\.\quad =-6+2(47)\\\\.\quad =-6+94\\\\.\quad =88`

`\text{Now use the sum formula:}\\\\S_n=(a_1+a_n)/(2)\cdot n\\\\\\S_(48)=(-6+88)/(2)\cdot 48\\\\\\.\quad =(82)/(2)\cdot 48\\\\\\.\quad =41\cdot 48\\\\.\quad =\large\boxed{1968}`

19 Answer: -116

Explanation:

`\{3, -2, -7, ...\}\\a_1=3,\ d=-5,\ n=8, \quad \text{solve for }a_(8)\\\\a_(n)=a_1+d(n-1)\\\\a_(8)=3-5(8-1)\\\\.\quad =3-5(7)\\\\.\quad =3-35\\\\.\quad =-32\\\\\text{Now use the sum formula:}\\\\S_8=(a_1+a_8)/(2)\cdot 8\\\\\\S_(8)=(3-32)/(2)\cdot 8\\\\\\.\quad =(-29)/(2)\cdot 8\\\\\\.\quad =-29\cdot 4\\\\.\quad =\large\boxed{-116}`