How many terms of the A.P. 22+26+30+… has the sum 400?

Explanation

To find how many terms of the arithmetic progression (A.P.) 22, 26, 30, ... have a sum of 400, we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
The first term \( A \) is 22. The common difference \( D \) can be calculated as:
\[ D = 26 - 22 = 4 \]

### Step 2: Use the formula for the sum of the first \( N \) terms of an A.P.
The formula for the sum of the first \( N \) terms \( S_N \) of an A.P. is given by:
\[ S_N = \frac{N}{2} \times (2A + (N - 1)D) \]
We know that \( S_N = 400 \), \( A = 22 \), and \( D = 4 \).

### Step 3: Substitute the known values into the formula
Substituting the values into the formula gives:
\[ 400 = \frac{N}{2} \times (2 \times 22 + (N - 1) \times 4) \]
This simplifies to:
\[ 400 = \frac{N}{2} \times (44 + 4(N - 1)) \]

### Step 4: Simplify the equation
Now, simplify the expression inside the parentheses:
\[ 400 = \frac{N}{2} \times (44 + 4N - 4) \]
\[ 400 = \frac{N}{2} \times (40 + 4N) \]
\[ 400 = \frac{N}{2} \times 4(N + 10) \]
\[ 400 = 2N(N + 10) \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 2N^2 + 20N - 400 = 0 \]
Dividing the entire equation by 2 simplifies it to:
\[ N^2 + 10N - 200 = 0 \]

### Step 6: Factor the quadratic equation
Now, we need to factor the quadratic equation:
\[ N^2 + 10N - 200 = 0 \]
This can be factored as:
\[ (N + 20)(N - 10) = 0 \]

### Step 7: Solve for \( N \)
Setting each factor to zero gives:
\[ N + 20 = 0 \quad \text{or} \quad N - 10 = 0 \]
Thus:
\[ N = -20 \quad \text{or} \quad N = 10 \]

### Step 8: Determine the valid solution
Since \( N \) represents the number of terms, it cannot be negative. Therefore, we reject \( N = -20 \) and accept:
\[ N = 10 \]

### Final Answer
The number of terms of the A.P. that sum to 400 is \( \boxed{10} \).

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