Find the first n-term sum Tₙ of bₙ.

01 / Standard#Your turn

a_n=2\cdot3^n,\quad b_n=\frac1{a_n}+\log_2a_n

Official paper · jm01-2023 · II.3(b) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Separate reciprocal and logarithmic contributions.

Hint 2

Use a geometric sum and 1+⋯+n.

Worked solution

Tₙ=(1−3⁻ⁿ)/4+n+n(n+1)log₂3/2.

Checks and common pitfalls: The first reciprocal term is 1/6, not 1/2.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Expand the logarithm.
$b_n=\frac1{2\cdot3^n}+1+n\log_2 3$
Sum each component.
$T_n=\frac14(1-3^{-n})+n+\frac{n(n+1)}2\log_2 3$

Think first. Reveal a hint when the class is ready.

2023 JM01