Question

The \(20^{th}\) term of the Arithmetic Progression \(10,\ 7,\ 4,\ \ldots\) is:

• A. \(-27\)
• B. \(-33\)
• C. \(-39\)
• D. \(-47\)

Hint:

For Arithmetic Progression: \[ a_n=a+(n-1)d \] If the sequence decreases, the common difference \(d\) will be negative.

Answer 1

The Correct Option is D

Concept: An Arithmetic Progression (A.P.) is a sequence in which the difference between consecutive terms remains constant. This constant difference is called the common difference and is denoted by \(d\). The formula for the \(n^{th}\) term of an Arithmetic Progression is: \[ a_n = a + (n-1)d \] where:
• \(a\) = first term
• \(d\) = common difference
• \(n\) = position of the required term
• \(a_n\) = \(n^{th}\) term

Step 1: Identifying the first term and common difference.
The given A.P. is: \[ 10,\ 7,\ 4,\ \ldots \] The first term is: \[ a=10 \] Now find the common difference: \[ d = 7-10 \] \[ d=-3 \] We can verify again: \[ 4-7=-3 \] Thus, the common difference is: \[ d=-3 \]

Step 2: Identifying the required term number.
We need to find the: \[ 20^{th} \] term. Thus: \[ n=20 \]

Step 3: Applying the \(n^{th}\) term formula.
Using: \[ a_n = a + (n-1)d \] Substitute the values: \[ a_{20}=10+(20-1)(-3) \] \[ =10+19(-3) \]

Step 4: Performing the calculations carefully.
Multiply: \[ 19 \times (-3)=-57 \] Now add: \[ 10+(-57) \] \[ =10-57 \] \[ =-47 \] Thus: \[ a_{20}=-47 \]

Step 5: Checking the options carefully.
Option (1): \[ -27 \] Incorrect. Option (2): \[ -33 \] Incorrect. Option (3): \[ -39 \] Incorrect. Option (4): \[ -47 \] Correct. Final Conclusion: The \(20^{th}\) term of the given Arithmetic Progression is: \[ \boxed{-47} \] Hence, the correct answer is option (4).