Solve the equation \(-4+(-1)+2+\cdots+x=437\).
\(x=50\)
Step 1: Identify the sequence.
The numbers are: \(-4, -1, 2, ... , x\).
This is an arithmetic progression (AP), because the difference between terms is constant.
Step 2: Find the first term (a) and common difference (d).
First term: \(a = -4\).
Second term: \(-1\).
Common difference: \(d = -1 - (-4) = 3\).
Step 3: General formula for the nth term.
In an AP, the nth term is:
\(T_n = a + (n-1) \cdot d\).
Here, the last term is \(x\). So:
\(x = -4 + (n-1) \cdot 3\).
Step 4: Rearrange to find n.
\(x + 4 = (n-1) \cdot 3\).
\(n - 1 = \dfrac{x+4}{3}\).
\(n = \dfrac{x+7}{3}\).
Step 5: Use the sum formula for an AP.
The sum of n terms is:
\(S_n = \dfrac{n}{2}(a + \text{last term})\).
Here, \(S_n = 437, a = -4, \text{last term} = x\).
So: \(437 = \dfrac{n}{2}(-4 + x)\).
Step 6: Substitute n.
\(437 = \dfrac{1}{2} \cdot \dfrac{x+7}{3} (x-4)\).
Simplify:
\(437 = \dfrac{(x+7)(x-4)}{6}\).
Step 7: Eliminate the fraction.
Multiply both sides by 6:
\(437 \times 6 = (x+7)(x-4)\).
\(2622 = x^2 + 3x - 28\).
\(x^2 + 3x - 2650 = 0\).
Step 8: Solve the quadratic equation.
Equation: \(x^2 + 3x - 2650 = 0\).
Using factorization (or quadratic formula), we get:
\(x = 50\) or \(x = -53\).
Step 9: Choose the valid answer.
The sequence starts from \(-4\) and increases by 3 each time. So a negative value like \(-53\) cannot appear in this sequence.
Therefore, the valid solution is:
\(x = 50\)