Find the 20th term of the AP 10,6,2,-2,-6,-10,……

Options

A. -66
B. 106
C. 96
D. none of these

Answer

To find the 20th term of the arithmetic progression (AP) given by the sequence 10, 6, 2, -2, -6, -10, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) is the first number in the sequence:
\[ a = 10 \]

Next, we need to find the common difference \( d \). The common difference is calculated by subtracting the first term from the second term:
\[ d = 6 - 10 = -4 \]

### Step 2: Use the formula for the nth term of an AP
The formula for the nth term \( T_n \) of an arithmetic progression is given by:
\[ T_n = a + (n - 1) \cdot d \]

### Step 3: Substitute n = 20 into the formula
We want to find the 20th term, so we substitute \( n = 20 \):
\[ T_{20} = 10 + (20 - 1) \cdot (-4) \]

### Step 4: Simplify the expression
Calculating \( (20 - 1) \):
\[ 20 - 1 = 19 \]

Now substituting this back into the equation:
\[ T_{20} = 10 + 19 \cdot (-4) \]

Calculating \( 19 \cdot (-4) \):
\[ 19 \cdot (-4) = -76 \]

Now substituting this value:
\[ T_{20} = 10 - 76 \]

### Step 5: Final calculation
Now, performing the final subtraction:
\[ T_{20} = 10 - 76 = -66 \]

Thus, the 20th term of the AP is:
\[ \boxed{-66} \]

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