10 Question MCQ (including Assertion)

Find the 5th term of the sequence in which the \(n^{th}\) term is \(t_n = n^2 - 2n + 3\).

A sequence is defined recursively by \(u_1=2\) and \(u_n=2u_{n-1}+1\) for \(n\ge 2\). What are its first four terms?

Assertion: If a sequence has the explicit rule \(t_n=4n+1\), then \(t_{100}\) can be found without calculating \(t_1,t_2,\ldots,t_{99}\).
Reason: An explicit rule gives each term only after all the preceding terms have been calculated.

In the growing pattern below, Stage \(1\) contains \(1\) square and each new stage adds \(4\) corner squares. How many small squares are present at Stage \(12\)?

Find the \(10\)th term of the GP \(5,25,125,\ldots\).

Identify the next term in the Virahānka-Fibonacci sequence: 1, 2, 3, 5, 8, 13, ...

An online channel has \(1200\) subscribers initially. Its subscriber count becomes \(1.5\) times the previous month’s count. What is the expected count after \(4\) months?

The Stage \(0\) Sierpiński square carpet has area \(1\) square unit. At each stage, \(\dfrac89\) of the previous red area remains. What is the red area at Stage \(3\)?

Is the sequence 1, -1, 1, -1, 1, ... a geometric progression? If so, what is the common ratio?