Question

use mathematical induction to prove the formula for every positive integer n.
3+7+11+15+...+(4n -1)=n(2n+1)

Answer 1

4n-1 does not equal n(2n+1)

Answer 2

for any n in N

Answer 3

let n = k
then if formula is true:
sum of k terms = k(2k + 1)

then sum of k+1 terms = k(2k + 1) + (4(k+1) - 1
= 2k^2 + k + 4k + 4 - 1
= 2k^2 + 5k + 3
= (2k + 3 )(k + 1 )

or (k + 1)(2(k + 1) + 1)

which has the same form as the formula for sum of k terms

so if its true for k terms then it is also true for k + 1 terms

it is true for first term so it must be true for 2nd term , then true for 3rd etc

so its true for all values of n

Answer 4

you sure you wrote ht eproblem right?

Answer 5

* (k + 1)(2(k + 1) + 1) has same form as k(2k + 1) with k being replaced by (k+1)

Answer 6

do you follow that garlucia?

Answer 7

so im confused but this does not seem to be true for any n, so what is the point of induction?

Answer 8

well its true for 1 term:

= 1(2*1 + 1) = 3

and if its true for kWh term it is also true for (k+1)th term because (k+1) replaces k in the formula

so true for term then it must be true for term , term 3, 5 and so on

thats method of induction

Answer 9

* true for term 1 , must be true for 2 , then true for term 3 and so on

Answer 10

but its not true for n = 3
(4*3-1) = 11
3(2*3+1) = 21

Answer 11

sum = n(2n + 1)

sum of 3 terms = 3(2(3) + 1) = 3* 7 = 21

- you are using the formula for nth term (4n-1) not formula for the sum of n terms

Answer 12

o snap sorry.....its been a long day

Answer 13

lol thats ok