The given sequence is an arithmetic sequence. To find the sum of this sequence, we need to use the formula for the sum of an arithmetic series:
| \(S_n = \frac{n}{2} (a + l)\) |
where:
In the given sequence: \(8 + 16 + 24 + 32 + \cdots + 80\), the first term \(a = 8\) and the last term \(l = 80\).
The common difference \(d\), found by subtracting the first term from the second term, is: \(d = 16 - 8 = 8\).
To find the number of terms \(n\), use the formula for the last term of an arithmetic sequence:
| \(l = a + (n-1)d\) |
Substituting the known values:
| \(80 = 8 + (n-1) \cdot 8\) |
Solving for \(n\):
| \(72 = (n-1) \cdot 8\) |
| \(n-1 = \frac{72}{8}\) |
| \(n-1 = 9\) |
| \(n = 10\) |
There are 10 terms in this arithmetic sequence.
Now apply the sum formula:
| \(S_n = \frac{10}{2}(8 + 80)\) |
| \(S_n = 5 \times 88\) |
| \(S_n = 440\) |
The value of the sum \(8 + 16 + 24 + 32 + \cdots + 80\) is 440.
Consider an A.P. $a_1,a_2,\ldots,a_n$; $a_1>0$. If $a_2-a_1=-\dfrac{3}{4}$, $a_n=\dfrac{1}{4}a_1$, and \[ \sum_{i=1}^{n} a_i=\frac{525}{2}, \] then $\sum_{i=1}^{17} a_i$ is equal to