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Prove by induction that the nth triangle number is given by n(n+1)/2

base case: (1 x 2)/2 = 1 as required inductive step: assuming statement holds for n=k, the (k+1)th triangle number is given by k(k+1)/2 + (k+1) by definition=(k^2+3k+2)/2=(k+1)(k+2)/2=(k+1)((k+1)+1)/2result ...
CB
Answered by Christopher B. • Maths tutor
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How would I go about drawing the graph of f(x) = sin(x)/(e^x) for -π≤x≤2π?

Drawing this graph uses the same techniques as with most other functions. Just from inspection, we can see that as x increases, e^x also increases, and so the amplitude of the function will decrease (and ten...
JZ
Answered by Jonah Z. • Maths tutor
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2 log(x + a) = log(16a^6) where a is a positive constant. How do I find x in terms of a?

The left hand side of the equation can be rewritten as log(x+a)^2, using the power rule for logs. We can then simplify log(x+a)^2=log(16a^6) to (x+a)^2=16a^6. Taking the square root of both sides, we find x+...
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Answered by Emily V. • Maths tutor
15460 Views

How do I solve quadratic equation by completing the square : X^2 - 4X = 5

We have X 2 - 4X = 5, write the expression on the left side as X 2 - 2•2X = 5. Then we would need extra term 4 to complete the square, so adding 4 on the both sides gives us, X 2 - 2•2X + 4 = 5 +4. Simplify ...
RK
Answered by Rabin K. • Maths tutor
4269 Views

Integrate cos(4x)+16x^3 with respect to x

This is a simple integration, integrating each individual term with respect to x.For the cos(4x), you should use 'integration by substitution' as it is a function of a function.cos(4x) = cos(u) and u = 4x wh...
AC
Answered by Aadil C. • Maths tutor
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