Find the 10th term from the end of the A.P. .
5+9+13...+93.

Answer

To find the 10th term from the end of the arithmetic progression (A.P.) given by the series 5, 9, 13, ..., 93, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term of the A.P. is:
- \( a = 5 \)

The common difference can be calculated as:
- \( d = 9 - 5 = 4 \)

### Step 2: Identify the last term (l)
The last term of the A.P. is:
- \( l = 93 \)

### Step 3: Determine the number of terms (n) in the A.P.
We can use the formula for the nth term of an A.P.:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ 93 = 5 + (n-1) \cdot 4 \]
Now, simplify and solve for \( n \):
\[ 93 - 5 = (n-1) \cdot 4 \]
\[ 88 = (n-1) \cdot 4 \]
\[ n - 1 = \frac{88}{4} \]
\[ n - 1 = 22 \]
\[ n = 23 \]

### Step 4: Find the 10th term from the end
The 10th term from the end is the same as the \( (n - 10 + 1) \)th term from the beginning:
\[ \text{Position from the start} = n - 10 + 1 = 23 - 10 + 1 = 14 \]

### Step 5: Calculate the 14th term
Using the formula for the nth term:
\[ T_n = a + (n-1)d \]
Substituting \( n = 14 \):
\[ T_{14} = 5 + (14 - 1) \cdot 4 \]
\[ T_{14} = 5 + 13 \cdot 4 \]
\[ T_{14} = 5 + 52 \]
\[ T_{14} = 57 \]

### Final Answer
The 10th term from the end of the A.P. is \( 57 \).

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