Find the 7th term from the end of A.P. 3+5+7+…+75.

Explanation

To find the 7th term from the end of the arithmetic progression (A.P.) given by the series 3, 5, 7, ..., 75, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) of the A.P. is 3, and the common difference \( d \) can be calculated as follows:
\[ d = 5 - 3 = 2 \]

### Step 2: Identify the last term (L)
The last term \( L \) of the A.P. is given as 75.

### Step 3: Determine the number of terms (n) in the A.P.
To find the number of terms \( n \) in the A.P., we can use the formula for the nth term of an A.P.:
\[ L = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 75 = 3 + (n - 1) \cdot 2 \]
Now, simplify the equation:
\[ 75 - 3 = (n - 1) \cdot 2 \]
\[ 72 = (n - 1) \cdot 2 \]
\[ n - 1 = \frac{72}{2} = 36 \]
\[ n = 36 + 1 = 37 \]
So, there are 37 terms in this A.P.

### Step 4: Find the 7th term from the end
To find the 7th term from the end, we can use the formula:
\[ \text{nth term from the end} = L - (n - k) \cdot d \]
where \( k \) is the position from the end (in this case, 7), \( L \) is the last term, \( n \) is the total number of terms, and \( d \) is the common difference.

Substituting the values:
\[ \text{7th term from the end} = 75 - (37 - 7) \cdot 2 \]
Calculating:
\[ = 75 - (30) \cdot 2 \]
\[ = 75 - 60 \]
\[ = 15 \]

### Final Answer
The 7th term from the end of the A.P. is **15**.

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