help finding the term of a sequence

Question

 

 

1.  help show the steps/formula to find the 100th term of 25, 10, -5 and -20.......

 

2.  help show teh steps/formula to find the tenth term of the sequence 5, -25, 125, -625......

 

 

3.  help show the steps/formula to find the sum of the first 20 terms of the series

 

          1 + 3 + 9 + 27 + .............

Answer 1

1. this is an arithmetic sequence

   use the formula x(n)=a+d(n-1) where a=first term, d=common difference, n=term number

   25, 10, -5, -20, ...

   x(100)=25+(-15)(100-1)

   x(100)=25+(-15)(99)

   x(100)=25+(-1485)

   x(100)=-1460

 

2. this is a geometric sequence

    use the formula A(n)=a(r)^n-1 where a=first term, r=common ratio, n=term number

    5, -25, 125, -625

    A(10)=5(-5)^10-1

    A(10)=5(-5)⁹

    A(10)=5(-5)(-5)(-5)(-5)(-5)(-5)(-5)(-5)(-5)
    A(10)=5(-5)(25)(25)(25)(25)

    A(10)=(-25)(625)(625)

    A(10)=(-25)(390,625)

    A(10)=-9,765,625

 

3.∑ 1+3+9+27+... +1,162,261,467 (20th term)

   ∑3^n-1

   use the formula S=a(rⁿ-1)/(r-1) where a=first term, r=common ratio, n=# of terms

   S=1(3²⁰-1)/(3-1)

   S=1(3,486,784,401-1)/(2)

   S=(3,486,784,400)/(2)

   S=1,743,392,200