Shift an exponential summation identity

Given ∑n=1N3nn+1(1−12n)=12(3N+1N+1−3)\sum_{n=1}^N\frac{3^n}{n+1}(1-\frac1{2n})=\frac12(\frac{3^{N+1}}{N+1}-3), deduce ∑n=1N3n(2n+1)(n+1)(n+2)\sum_{n=1}^N\frac{3^n(2n+1)}{(n+1)(n+2)}, leaving the answer as a(3N+2N+2−b)a(\frac{3^{N+2}}{N+2}-b) where a,ba,b are constants to be determined. marks

[modified 2020/ACJC/Prelim/I/5]

Solution

Match the summands

Find the shift

Given ∑n=1N3nn+1(1−12n)=12(3N+1N+1−3)\sum_{n=1}^N\frac{3^n}{n+1}\left(1-\frac1{2n}\right)=\frac12\left(\frac{3^{N+1}}{N+1}-3\right), relate the summand of ∑n=1N3n(2n+1)(n+1)(n+2)\sum_{n=1}^N\frac{3^n(2n+1)}{(n+1)(n+2)} to the given summand Tn=3n(2n−1)2n(n+1)T_n=\frac{3^n(2n-1)}{2n(n+1)}.

Hint

  1. Rewrite the given summand as one fraction.

  2. Try replacing nn by n+1n+1 in it.

Answer · Match the summands

The target summand is 23Tn+1\dfrac23T_{n+1}.

Evaluate the sum

Sum with shifted limits

Deduce ∑n=1N3n(2n+1)(n+1)(n+2)\sum_{n=1}^N\frac{3^n(2n+1)}{(n+1)(n+2)} in the form a(3N+2N+2−b)a\left(\frac{3^{N+2}}{N+2}-b\right).

Explore this idea

Hint

  1. Use 23Tn+1\frac23T_{n+1} and shift the limits to 2,…,N+12,\dots,N+1.

  2. Subtract T1T_1 to start from 11.

Answer · Evaluate the sum

13(3N+2N+2−92)\frac13(\frac{3^{N+2}}{N+2}-\frac92); a=1/3,b=9/2a=1/3,b=9/2.

Explore the idea

Growing terms, and a tidy closed form

Each bar is 3n(2n+1)(n+1)(n+2)\frac{3^n(2n+1)}{(n+1)(n+2)}. Move the upper limit and compare the total with 13(3N+2N+2−92)\frac13\left(\frac{3^{N+2}}{N+2}-\frac92\right).

Upper limitr = 1 to 3
Try an example
Σ 3ⁿ(2n+1)/((n+1)(n+2)), r = 1 to 3
14.7
Number of terms
3
⅓(3^(N+2)/(N+2) − 9/2)
14.7
matches the direct total
Key idea

A summand that is a constant multiple of a shifted given summand has a total that is the same multiple of a shifted given total, minus the terms the shift skips.

Ideas to try
  1. At N=1N=1 the only term is 3⋅32⋅3=1.5\frac{3\cdot3}{2\cdot3}=1.5, and the formula gives 13(273−92)=1.5\frac13\left(\frac{27}3-\frac92\right)=1.5.
  2. Watch the exponential growth: each bar is nearly three times the last.
Takeaway

Shift, scale, trim

To use a given sum, shift the index to match, scale by the constant ratio, and trim the terms that fall outside the given range.