Find the number of terms in this AP

Question

The AP is defined as follows:

$$ 7 + 9 + ... + (2n + 1) $$

If I remember correctly the answer given was $n-2$.

But I thought that since $n$ was defined as the number of terms in the progression, the answer should always be $n$.

How can the answer be $n - 2$ (or in fact anything other than $n$).

Thanks in advance for the help

Answer 1

Find out the common difference denoted by $d_1$.

Let the first term be represented by $A$ and let the last term be represented by $L$.

Calculate number of terms $T$ using the formula : $T = \frac{L-A}{d_1} + 1$. We thus get $T = n-2$ using $L = 2n+1$ and $A= 7$.

---

As to why the answer is $n-2$, one can explain it in terms of the direct consequence of the formula. It is not always necessary that the number of terms in a sequence should be $n$. Your sequence is that of the odd numbers starting from $7$ and not from $3$. The latter case would have given you $n$ terms. Hope it helps.

Answer 2

$$ 1+\frac{1}{2}((2n+1)-7)=n-2. $$

Answer 3

Suppose that $m$ is then number of terms in the sequence. Denote the first term by $a_1$, and then you can see that

$$ a_m=5+2m $$

then $a_m=2n+1$, $5+2m=2n+1$ so $2m=2n-4$ whence you get $m=n-2$

Answer 4

$$a_n= 2n+1$$ $$a_1 = 3$$ $$a_n=7$$ $$2n+1=7$$ $$n=3$$ $$a_3=7$$

So series will become 3,5,7,9,…

So before 7 there are 2 terms which are not included in your series. So there are total $n-2$ terms