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Evaluating a constant sigma sum

Consider the constant function f(x)=5f(x) = 5, and suppose we partition an interval [a,b][a,b] into nn equal subintervals.

Evaluate the sum ∑i=1n5\sum_{i=1}^{n} 5 in terms of nn, and explain what this sum represents geometrically if each term 55 is thought of as the height f(xi∗)=5f(x_i^*) = 5 of a rectangle of width Δx\Delta x in a Riemann sum (so the full Riemann sum would be ∑i=1n5Δx\sum_{i=1}^{n} 5 \Delta x). State the value of ∑i=1n5\sum_{i=1}^{n} 5 and briefly describe the geometric meaning.

Hint

Since every term in the sum is the same constant 55, adding it up nn times is just repeated addition — no summation formulas for ii or i2i^2 are needed here.

Step-by-step solution

Since the summand does not depend on ii, every one of the nn terms equals 55. So the sum is just 55 added to itself nn times: ∑i=1n5=5+5+⋯+5⏟n terms=5n.\sum_{i=1}^{n} 5 = \underbrace{5+5+\cdots+5}_{n \text{ terms}} = 5n.

Geometrically, if this sum is scaled by Δx\Delta x to form a Riemann sum ∑i=1n5Δx=5nΔx=5(b−a)\sum_{i=1}^{n} 5\Delta x = 5n\Delta x = 5(b-a), each term 5Δx5\Delta x is the area of a thin rectangle of height 55 and width Δx\Delta x. Since f(x)=5f(x)=5 is constant, all rectangles have the same height, so the sum of their areas exactly equals the area of the rectangle of height 55 over the whole interval [a,b][a,b] — no approximation error occurs.

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