Prove that 8 times any triangle number is always 1 less than a square number

A triangle number is a number such that it is the sum of n consecutive integers, starting from 0. Eg 1, 1+2, 1+2+3... are the first 3 triangle numbers. The formula for the nth triangle number is well-known at A-level and is (1/2)(n)(n+1); the formula for the sum of the first n integers. To answer the question we must show that 8N+1, where N is any triangle number, is a perfect square.
8N+1 = 8(1/2)(n)(n+1)+1 = 4n(n+1)+1=4n^2 + 4n + 1This is a perfect square, as we can rewrite it as:(2n+1)^2with it's root of 2n+1 being an integer. Therefore we have shown that 8N+1 is a perfect square, hence the result has been proved.

Answered by Maths tutor

8485 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Find the area bounded by the curve y=(sin(x))^2 and the x-axis, between the points x=0 and x=pi/2


Differentiate with respect to x, y = (x^3)*ln(2x)


Find the integral of (sinxcos^2x) dx


Find the antiderivative of the function f(x)=cos(2x)+5.


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences