IF the 10th term of the sequence a, a-b, a-2b,a-3b …….is 20 and the 20th term is 10 then the xth term of the series is

Options

A. 10-x
B. 20-x
C. 29-x
D. 30-x

Answer

To solve the problem step-by-step, we will first identify the terms of the sequence and then use the information given to find the required term.

### Step 1: Identify the terms of the sequence
The sequence is given as:
- First term: \( a \)
- Second term: \( a - b \)
- Third term: \( a - 2b \)
- Fourth term: \( a - 3b \)
- ...
- \( n \)-th term: \( a - (n-1)b \)

### Step 2: Write the equations for the 10th and 20th terms
From the problem, we know:
- The 10th term is \( a - 9b = 20 \) (1)
- The 20th term is \( a - 19b = 10 \) (2)

### Step 3: Set up the equations
From equation (1):
\[ a - 9b = 20 \]

From equation (2):
\[ a - 19b = 10 \]

### Step 4: Solve the equations
Now, we will solve these two equations simultaneously.

Subtract equation (1) from equation (2):
\[ (a - 19b) - (a - 9b) = 10 - 20 \]
This simplifies to:
\[ -19b + 9b = -10 \]
\[ -10b = -10 \]
\[ b = 1 \]

### Step 5: Substitute \( b \) back to find \( a \)
Now that we have \( b \), we can substitute it back into one of the original equations to find \( a \). Let's use equation (1):
\[ a - 9(1) = 20 \]
\[ a - 9 = 20 \]
\[ a = 29 \]

### Step 6: Find the \( x \)-th term of the sequence
The \( x \)-th term of the sequence is given by:
\[ T_x = a - (x-1)b \]
Substituting the values of \( a \) and \( b \):
\[ T_x = 29 - (x-1)(1) \]
\[ T_x = 29 - (x - 1) \]
\[ T_x = 29 - x + 1 \]
\[ T_x = 30 - x \]

### Final Answer
The \( x \)-th term of the sequence is:
\[ T_x = 30 - x \]