Question

A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that each of these statements is true. Show your work.

Sn: 2 + 5 + 8 + . . . + ( 3n - 1) = n(1 + 3n)/2
my answer:
S1: 2+5+8+(3*1-1)=1(1+3*1)/2 =2
S2: 2+5+8+(3*2-1)=2(1+3*2)/2 =4
S3: 2+5+8+(3*3-1)=3(1+3*3)/2 =5
All the statements are true.
my teacher said:
n is the sum of n terms given by the formula. Here Sn = n(1+3n)/2.
Use the expression for the term to find each successive term and then show that the sum formula
works for these n values, i.e. n=1, 2 & 3.
@mathmath333

Answer 1

i already did what she is asking n=1,2,3 then whats left ? |@mathmath333

Answer 2

well i asked u before does ur teacher asked for proof by induction or not

Answer 3

this is anathor Question

Answer 4

i dont think so she said: n is the sum of n terms given by the formula. Here Sn = n(1+3n)/2.
Use the expression for the term to find each successive term and then show that the sum formula
works for these n values, i.e. n=1, 2 & 3.

Answer 5

\[n(1+3n)/2\\for~n=1\\1(1+3\times 1)/2=2\\for~n=2\\2(1+3\times 2)/2=7\\for~n=3\\3(1+3\times 3)/2=15\]

Answer 6

please explain a bit

Answer 7

@mathmath333

Answer 8

u have to calculate the sum upto the nth term

suppose u have to calculate upto 3 terms

so series would be
\(2+5+8=15\)

but u can also use the formula

\(\dfrac{n(1+3n)}{2}\\
=3\dfrac{(1+3\times 3)}{2}\\=15\)

Answer 9

both of ur answer is same just deifferent way

Answer 10

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work.

2 is a factor of n2 - n + 2

Answer 11

Steps for a proof by induction:- 1 First prove it is true for n=1
2.Assume it is true for n=k 3. Show that it is true for n= (k+1) This may require you to do some
algebraic manipulations but once you complete the last step your proof is done.
You will find if you remove the parentheses you can simplify into the original expression for k +
another term that is divisible by 2.
@mathmath333 @hartnn please help

Answer 12

thats all i could do:
S(n) : 2 is a factor of n^2
S(1) : 2 is a factor of 1^2
S(2) : 2 is a factor of 2^2
are S(1) and S(2) true

Answer 13

@hartnn dear u there please help

Answer 14

\(\large\tt \begin{align} \color{black}{\color{red}{1.)}\\~\\
for~n=1\\~\\
n^2 - n + 2\\~\\
=(1)^2 - 1 + 2\\~\\
=2\\~\\
\text{hence it is true for }~n=1\\~\\
\color{red}{2.)}\\~\\
\text{assume that it is true for }~n=k\\~\\
\text{so it is true for }\\~\\
k^2 - k + 2\\~\\
\text{now we have to prove it is true for }~n=k+1\\~\\
(k+1)^2 -( k+1) + 2\\~\\
=k^2+2k+1 - k-1 + 2\\~\\
=(k^2 - k+ 2)+2k\\~\\
(k^2 - k+ 2) ~\text{ is true as we assumed }\\~\\
\text{and 2k is a factor of 2 }\\~\\
\text{PROVED! }\\~\\
}\end{align}\)

Answer 15

thanks =)