Question

Consider the sequence of integers: 122333444455555 ............. where n appears n times. The t1h 9te9r4m is

Answer 1

the sequence of integers; 19999999994444

Answer 2

The correct question: Consider the sequence of integers: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5,...…. where n appears n times. The 100th term of this sequence is _______.

The 100th term of the given sequence is 14.

Given:

The given sequence = 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5,...….

To Find:

We have to find the 100th term of the given sequence of integers.

Solution:

Let n be the 100th term. Each number of the given sequence is repeated its own times. Therefore, the number of terms can be given by:

$\frac{n (n+1)}{2} = 100$

$i.e., n (n+1) = 200$

Or, $n^2 + n - 200 = 0$

This is in the form of a second degree quadratic equation. The value of n is then given by,

$n = \frac{-1 ± \sqrt{(1)^2 + (4 × 1 × 200)} }{2 × 1}$  $= 13.65$ ≅ $14.$

∴, The 100th term = 14.

Hence, the 100th term of the given sequence is 14.

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