An integer upper limit in an inequality

Find the set of values of aa for which ∑r=3a(2r+a)>a2\sum_{r=3}^{a}(2r+a)>a^2. marks

[2015/TJC/MCT/1]

Solution

Evaluate the sum

Sum an AP written in sigma notation

Evaluate ∑r=3a(2r+a)\sum_{r=3}^{a}(2r+a) in terms of aa, for integer a≥3a\ge3.

Explore this idea

Hint

  1. Write out the first terms: rr varies, aa is a constant.

  2. Count the terms: upper minus lower plus one.

  3. Use n2(first+last)\tfrac n2(\text{first}+\text{last}).

Answer · Evaluate the sum

∑r=3a(2r+a)=(a−2)(4a+6)2=2a2−a−6\sum_{r=3}^{a}(2r+a)=\dfrac{(a-2)(4a+6)}2=2a^2-a-6.

Explore the idea

The terms 2r + 5, summed from r = 3

This example fixes a=5a=5, so the terms are 2r+52r+5 and the upper limit is 55. Drag the upper limit to see the AP grow.

Upper limitr = 3 to 5
Try an example
Σ 2r + 5, r = 3 to 5
39
Number of terms
3
AP formula
39
matches the direct total
Key idea

A sum with a linear term is an arithmetic progression. Its value is the number of terms times the average of the first and last.

Ideas to try
  1. With the upper limit at 55 the terms are 11, 13, 1511,\ 13,\ 15 and the sum is 39=2(25)−5−639=2(25)-5-6.
  2. Switch to the running total to watch the AP add a slightly bigger step each time.
Solve the inequality

Solve a quadratic inequality and respect the limits

Find the set of values of aa for which ∑r=3a(2r+a)>a2\sum_{r=3}^{a}(2r+a)>a^2.

Explore this idea

Hint

  1. Substitute the sum and rearrange to 00.

  2. Solve the quadratic inequality.

  3. Reject values that make the upper limit smaller than the lower limit.

Answer · Solve the inequality

a∈Za\in\mathbb Z, a≥4a\ge4.

The algebraic branch a<−2 is incompatible with the finite-sum bounds.

Explore the idea

When does the sum overtake a²?

The sum ∑r=3a(2r+a)=2a2−a−6\sum_{r=3}^{a}(2r+a)=2a^2-a-6 against a2a^2, for each integer aa. The gap is a2−a−6a^2-a-6.

Value of aa = 4
Try an example
Σ(2r + a) at a = 4
22
a² at a = 4
16
gap = Σ − a² at a = 4
6
Key idea

The root a=−2a=-2 is not admissible, and the root a=3a=3 is excluded by the strict inequality, so the first value that works is 44.

Ideas to try
  1. Step to a=3a=3: the gap is zero, so the strict inequality fails.
  2. Move to a=4a=4. The gap becomes 66 and keeps growing.
Takeaway

Limits are part of the question

A finite sum only exists when its upper limit is at least its lower limit. After solving the algebra, check it against that rule.