A polynomial term and a middle block

Given ∑r=1nr2=n6(n+1)(2n+1)\sum_{r=1}^nr^2=\frac n6(n+1)(2n+1), show ∑r=1n(2r+1)(2r+3)=n3(4n2+18n+23)\sum_{r=1}^n(2r+1)(2r+3)=\frac n3(4n^2+18n+23). Hence find the exact sum of the 25th to the 75th term of the series. marks

[2023/DHS/Prelim/II/2]

Solution

Show the formula

Expand and sum

Given ∑r=1nr2=n6(n+1)(2n+1)\sum_{r=1}^nr^2=\frac n6(n+1)(2n+1), show that ∑r=1n(2r+1)(2r+3)=n3(4n2+18n+23)\sum_{r=1}^n(2r+1)(2r+3)=\frac n3(4n^2+18n+23).

Explore this idea

Hint

  1. Expand the product.

  2. Sum 4r24r^2 with the given result and 8r+38r+3 as an AP.

Answer · Show the formula

Sn=n(4n2+18n+23)/3S_n=n(4n^2+18n+23)/3.

Explore the idea

Products of consecutive odd numbers

The bars are 3⋅5, 5⋅7, 7⋅9,…3\cdot5,\ 5\cdot7,\ 7\cdot9,\dots. Move the upper limit and compare with the formula.

Upper limitr = 1 to 4
Try an example
Σ (2r+1)(2r+3), r = 1 to 4
212
Number of terms
4
n(4n²+18n+23)/3
212
matches the direct total
Key idea

Quadratic terms always give a cubic total.

Ideas to try
  1. At n=1n=1 the formula gives 4+18+233=15=3×5\frac{4+18+23}3=15=3\times5.
  2. Check n=2n=2: 15+35=5015+35=50 and 2(16+36+23)3=50\frac{2(16+36+23)}3=50.
The 25th to 75th terms

Block as a difference

Hence find the exact sum of the 25th to the 75th terms of the series.

Explore this idea

Hint

  1. The kkth term is the term with r=kr=k.

  2. Compute S75−S24S_{75}-S_{24}.

Answer · The 25th to 75th terms

574753574753.

Explore the idea

The same subtraction on a smaller block

Select a block of terms and compare its direct total with Supper−Slower−1S_{\text{upper}}-S_{\text{lower}-1}, the method used for the 25th to 75th terms.

Limits of the sumr = 5 to 9
Try an example
Σ (2r+1)(2r+3), r = 5 to 9
1315
Number of terms
5
S(upper) − S(lower − 1)
1315
matches the direct total
Key idea

Subtract Sa−1S_{a-1}, not SaS_a: the first term of the block must stay.

Ideas to try
  1. Pick terms 55 to 99. The grey bars on the left are the four terms that S4S_{4} removes.
  2. Move the lower limit one step right and see the difference formula lose exactly one term.
Takeaway

Total minus the start

For a middle block, find the total up to the end and take away the total just before the start.