2 - 1 series and induction

Question 1

proving by induction the nth term
eg u1 = 2
u n+1 = 3 un + 2
prove un = 3^n -1

Answer 1

when n = 1
u1 = 3-1 = 2
assume when n = k
when n = k + 1
uk+1 = 3uk + 2
3(3^k -1) + 2
= 3^k+1 -1

Question 2

if the sequence determines on 2 previous terms

Answer 2

can still prove by induction but show it works for 2 starting values of n

Question 3

prove by induction series

Answer 3

same thing
n = 1
assume for n = k
let n = k + 1
but sum to k+1 = sum to k + uk+1

Question 4

induction conclusion

Question 5

sum of r from n to 1

Answer 5

1/2 n(n+1)

Question 6

sum of c from r=1 to n when c is constant

Answer 6

nc

Question 7

method of differences

Answer 7

if a series is in the form f(r+1) - f(r)
the sum = f(n+1) - f(1)
write it out for the first few terms and last few to see the cancellation
usually have to split up by partial fractions

Question 8

sum yo infinity of a convergent series

Answer 8

find the sum of the first n terms and consider what happens as n gets large