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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

Sam Houston
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    More often than not, for formulas involving arithmetic sequences (e.g. generic term, sums) $n$ happens to be the number of terms. But it is not required to be, it's often just a variable used to index terms. – pjs36 Dec 24 '16 at 09:43

4 Answers4

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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.

  • I think this is a good answer. I would like to add fact, this question should probably be formed using different letter, like 'k'. This is similar confusion when we want for example calculate area of square equal to d^2 where d usually means diagonal. – LAAE Dec 24 '16 at 11:04
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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 $m=n-2$

user160738
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$$a_n= 2n+1$$ $$a_1 = 3$$ $$a_n=7$$ $$2n+1=7$$ $$n=3$$ $$a_3=7$$

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

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

Paolo Leonetti
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