The general term of an arithmetic sequence is $T_n = 15 - 2n$, where $n \in \mathbb{N}$ - Leaving Cert Mathematics - Question a) - 2014

To find the first negative term, set $T_n < 0$:

$15 - 2n < 0$

Rearranging gives:

$2n > 15$

Dividing both sides by 2 results in:

$n > 7.5$

The smallest integer $n$ that satisfies this inequality is 8. Thus, the first negative term occurs at $T_8$.

The sum of the first $n$ terms of an arithmetic sequence is given by:

$S_n = \frac{n}{2} (T_1 + T_n)$

First, we need to calculate $T_n$:

$T_n = 15 - 2n$

Thus,

$S_n = \frac{n}{2} \left( 13 + (15 - 2n) \right) = \frac{n}{2} (28 - 2n) = n(14 - n)$

Set the sum equal to zero:

$S_n = n(14 - n) = 0$

This gives us two possible solutions:

• $n = 0$ (not applicable since $n \in \mathbb{N}$)
• $14 - n = 0 \Rightarrow n = 14$.

Therefore, the value of $n$ for which the sum is 0 is $n = 14$.