Question

Consider the sequence a1 = 101, a2 = 10101, a3 =1010101 and so on

Answer 1

Answer:

A closed formula is

$a_n = \frac{100^{n+1}-1}{99}$

Step-by-step explanation:

Notice a1 = 100 + 1

a2 = 100² + 100 + 1

a3 = 100³ + 100² + 100 + 1

and so on.

In general, each term is obtained by multiplying the previous one by 100 and then adding 1.

This might look more familiar written like this:

$a_n = 1 + x + x^2 +\dots+ x^n$

where in our case we have x = 100.

This is a sum of terms from a geometric series, and the formula for such a sum is

$1 + x + x^2 +\dots+ x^n = \frac{x^{n+1} - 1}{x-1}$

Since we have x = 100 in our case, we just put 100 into this formula and that gives the formula for the terms in our sequence.