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Worked solution

Rearrange a recurrence before generating terms

Given un+1−3un=4u_{n+1}-3u_n=4, u1=5u_1=5, evaluate u2,u3,u4u_2,u_3,u_4.

CJC · Sequences and Series · Example 4

Answer

u2=19u_2=19, u3=61u_3=61, u4=187u_4=187.

Compare your reasoning

Method 1: isolate the next term

un+1−3un=4⟹un+1=3un+4u_{n+1}-3u_n=4\quad\Longrightarrow\quad u_{n+1}=3u_n+4

Retain the given initial value while rearranging.

Substitute the preceding values

u2=3u1+4=3(5)+4=19,u3=3u2+4=3(19)+4=61,u4=3u3+4=3(61)+4=187.\begin{aligned}u_2&=3u_1+4=3(5)+4=19,\\u_3&=3u_2+4=3(19)+4=61,\\u_4&=3u_3+4=3(61)+4=187.\end{aligned}

All three substitutions are printed in both copies.

Method 2: use the next-term calculator type

On the source device in SEQ mode: (1) press y=, use Up/Right to highlight TYPE SEQ(n+1), then Enter; (2) set nMin=1; (3) use 2nd 7 for u, enter 3u(n)+4 with X,T,θ,n for n, and set u(1)=5; (4) press 2nd graph and inspect with Up/Down. The source table starts 5,19,61,187,565,1699,5101,15307,45925,137779,413341.

How did this question go?

Common mistakes

Match the calculator TYPE to the written recurrence. Typing n+1 on the wrong side changes the indexing.