what is 70+68+66+...+4+2?
what is the shortcut
2(1+2+3+...+35)
Recall that 1+2+3+...+n = n(n+1)/2
So now we have
2 * 35*36/2 = 1260
thanks!
To find the sum of a series of numbers, you can use the formula for the sum of an arithmetic series:
Sn = (n/2)(a + l)
Where:
- Sn is the sum of the series,
- n is the number of terms in the series,
- a is the first term, and
- l is the last term.
In this case, the first term is 70, the last term is 2, and we need to find the sum of numbers that decrease by 2 with each term.
To find the number of terms, we can use the formula for an arithmetic sequence:
aN = a + (n - 1)d
Where:
- aN is the nth term of the sequence,
- a is the first term,
- n is the number of terms, and
- d is the common difference.
In this case, aN = 2, a = 70, and d = -2. We need to solve for n:
2 = 70 + (n - 1)(-2)
Simplifying the equation:
2 = 70 - 2n + 2
2n = 70
n = 35
Now, we have all the information to find the sum of the series:
Sn = (35/2)(70 + 2)
Sn = (35/2)(72)
Sn = 1260
Therefore, the sum of the series 70+68+66+...+4+2 is 1260.
The shortcut for finding the sum of an arithmetic series is to use the formula:
Sn = (n/2)(a + l)
By plugging in the values directly into the formula, you can quickly find the sum without having to manually add up each term.