+753 1.Part (a): Find the sum \(a + (a + 1) + (a + 2) + \dots + (a + n - 1)\)in terms of a and n. Part (b): Find all pairs of positive integers (a,n) such that \(n \ge 2\) and \([a + (a + 1) + (a + 2) + \dots + (a + n - 1) = 100.\)
2. Find x such that \(\lceil x \rceil + x = \dfrac{23}{7}\) . Express x as a common fraction.
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2)
Solve for x over the real numbers:
x + abs(x) = 23/7
Hint: | Isolate terms with abs(x) to the left hand side.
Subtract x from both sides:
abs(x) = 23/7 - x
Hint: | Eliminate the absolute value.
Split the equation into two possible cases:
x = 23/7 - x or x = x - 23/7
Hint: | Look at the first equation: Isolate x to the left-hand side.
Add x to both sides:
2 x = 23/7 or x = x - 23/7
Hint: | Solve for x.
Divide both sides by 2:
x = 23/14 or x = x - 23/7
Hint: | Look at the second equation: Isolate x to the left-hand side.
Subtract x from both sides:
x = 23/14 or 0 = -23/7
Hint: | Look for a false statement.
0 = -23/7 is trivially false:
x = 23/14
1)
I can only see two pairs of a, n that satisfy the given conditions:
9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 = 100, where a = 9 and n = 8
18 + 19 + 20 + 21 + 22 = 100, where a = 18 and n = 5
2)
x + abs(x) = 23/7, solve for x
Split it into 2 equations as follows:
+x + x = 23/7.............(1)
- x + x = 23/7.............(2)
In equation (1) you have:
2x = 23/7 divide both sides by 2
x = (23/7) / 2
x = 23/7 x 1/2
x = 23/14
In equation (2) you have:
-x + x = 23/7
0 = 23/7 Discard this, since it is false.
