Consider the sequence of sums 1,1+2,1+2+3,...

Consider the sequence of sums $1,1+2,1+2+3$, $1+2+3+4,1+2+3+4+5, \ldots$ (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 $k$ th term of this sequence of sums is greater than 1000 .

Consider the sequence of sums $1,1+2,1+2+3$, $1+2+3+4,1+2+3+4+5, \ldots$
(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 $k$ th term of this sequence of sums is greater than 1000 .

Key Concepts

Solving Quadratic Inequalities

To determine the point in the sequence beyond which every term exceeds a particular value (such as 1000), one must solve a quadratic inequality based on the triangular numbers formula. This involves formulating and solving an inequality like n(n+1)/2 > 1000, which requires knowledge of quadratic equations and their roots, thereby linking algebraic methods to the analysis of sequence behavior.

Quadratic Growth

The quadratic nature of the triangular numbers formula means that as n increases, the growth rate of the sum is proportional to n^2. This growth behavior is significantly faster than linear growth, which is crucial for understanding why the sequence becomes very large for sufficiently high values of n.

Triangular Numbers

The sequence of sums in the problem represents triangular numbers, which are formed by summing the first n natural numbers. The nth triangular number is given by the formula T(n) = n(n+1)/2. This concept is central to understanding the structure and behavior of the sequence, as it reveals that each term is not arbitrary but follows a predictable and precise mathematical formula.

Divergence of Sequences

A key concept here is the behavior of the sequence as the index increases. Since the formula for each term is T(n) = n(n+1)/2, which is a quadratic function of n, the terms of the sequence grow without bound as n increases. This implies that not only do the terms become larger and larger, but they also tend to infinity, a concept known as divergence.

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