viktorija wrote:

If sequence T is defined for all positive integers n such that \(t_{n +1} = t_{n} + n\), and \(t_3 = 14\), what is \(t_{20}\) ?

A. 101

B. 187

C. 191

D. 201

E. 251

Source : Manhattan Advanced Quant Question No. 8

OFFICIAL SOLUTION

First, we can write a few of the terms, noting the relationship of each term to the previous one:

\(t_3 = 14\)

\(t_4 = 14 + 3\)

\(t_5 = 14 + 3 + 4\)

\(t_6 = 14 + 3 + 4 + 5\)

…

\(t_{20} = 14 + (3 + 4 + 5 + … + 18 + 19)\)

To evaluate \(t_{20}\) , we need to compute the sum contained in parentheses above.

We can use the rule that (sum of a set of consecutive integers) = (middle term) × (number of terms).

The middle term can be found by taking the average of the two extreme terms 3 and 19 to get 11. The number of terms is \(19 – 3 + 1 = 17\).

Now we can compute \(11 × 17 = 187\).

Finally, we have \(t_{20} = 14 + (3 + 4 + 5 + … + 18 + 19) = 14 + 187 = 201\).

The correct answer is D.

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