Understanding the Arithmetic Series

The given series is $1+3+5+7+...$. This is an arithmetic series where:

• The first term ($a$) is $1$.
• The common difference ($d$) is $3 - 1 = 2$.

We need to find the number of terms ($n$) such that the sum of these terms ($S_n$) equals $12345678987654321$.

Calculating the Sum of the Series

The formula for the sum of the first $n$ terms of an arithmetic series is:

$S_n = \frac{n}{2} [2a + (n-1)d]$

Substitute the values $a=1$ and $d=2$ into the formula:

$S_n = \frac{n}{2} [2(1) + (n-1)2]$

$S_n = \frac{n}{2} [2 + 2n - 2]$

$S_n = \frac{n}{2} [2n]$

$S_n = n^2$

This shows that the sum of the first $n$ odd numbers is equal to the square of $n$. In this specific series, the sum of the first $n$ terms is $n^2$.

Finding the Number of Terms

We are given that the sum $S_n = 12345678987654321$. Therefore, we have the equation:

$n^2 = 12345678987654321$

To find $n$, we need to calculate the square root of $12345678987654321$.

Let's examine the pattern of squares of numbers consisting of ones:

By observing the pattern, we can see that the square root of $12345678987654321$ is $111111111$.

So, $n = \sqrt{12345678987654321} = 111111111$

Conclusion

The number of terms of the series $1+3+5+7+...$ that amount to a sum equal to $12345678987654321$ is $111111111$.