Q. 11 - Calculus

Question

Consider the sequence of sums 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, 1 + 2 + 3 + 4 + 5, .... (a) What happens to the terms of this sequence of sums as k gets larger and larger? (b) Find a sufficiently large value of k that will guarantee that every term past the kth term of this sequence of sums is greater than 1000. 

Step-by-Step Solution

Answer

(a) $\underset{k\to \mathrm{\infty }}{lim} \frac{k(k+1)}{2}=\mathrm{\infty }$

(b) 990

Part (a) Step 1. Given information

Given is the sequence 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, 1 + 2 + 3 + 4 + 5, .... 

We have to explain What happens to the terms of this sequence as k gets larger and larger and find a sufficiently large value of k that will guarantee that every term past the kth term of this sequence of sums is greater than 1000. 

Part (a) Step 2. Terms of the sequence

From the sequence, we see that, as k gets larger and large, the terms get larger and larger.

Therefore, in limit expression it can be written as below:

$\underset{k\to \mathrm{\infty }}{lim} \frac{k(k+1)}{2}=\mathrm{\infty }$

Part (b) Step 1. Value of k

For every k>44, the terms will be :

$\begin{array}{r}\frac{44(44+1)}{2}\\ =22\left(45\right)\\ =990\end{array}$