Prove that sum(k) from 0 to n is n(n+1)/2, by induction

Proof by induction involves making an assumption, and using that assumption to prove that the consecutive case follows the pattern. 

The key to this is realising that most questions follow the same structure, usually involving rearranging algebra. Remember to try to see where you can use the induction step, and how you can rearrange it to make it clear how the induction step fits in. Just keep calm, write out every step carefully, and the answers will follow.

Base case: for k=1, sum(0+1) = 1 and 1(1+1)/2 = 1, and we have shown that the claim is true in this case.

Hypothesis: suppose the claim is true for k=n

Induction step: for k=n+1 , take the sum:

sum(k) [0--n+1] = sum(k)[0--n] + n+1 = n(n+1)/2 +n+1 = (n2+n)/2 + (2n+2)/2 = (n2+3n+2)/2

= (n+1)(n+2)/2 and we have shown the claim

Conclusion: As the claim is true for 0 and 1, and we have shown it to be true if it is true for n=k, by induction we have proved it true for all n in the natural numbers.

NR
Answered by Nadine R. Further Mathematics tutor

7802 Views

See similar Further Mathematics A Level tutors

Related Further Mathematics A Level answers

All answers ▸

It is given that f(x)=(x^2 +9x)/((x-1)(x^2 +9)). (i) Express f(x) in partial fractions. (ii) Hence find the integral of f(x) with respect to x.


Let A, B and C be nxn matrices such that A=BC-CB. Show that the trace of A (denoted Tr(A)) is 0, where the trace of an nxn matrix is defined as the sum of the entries along the leading diagonal.


The ODE mx'' + cx' + kx = 0 is used to model a damped mass-spring system, where m is the mass, c is the damping constant and k is the spring constant. Describe and explain the behaviour of the system for the cases: (a) c^2>4mk; (b) c^2=4mk; (c) c^2<4mk.


How do I know when I should be using the Poisson distribution?


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning