Question

The \( 30^{th} \) term of the arithmetic progression \( 10, 7, 4, \dots \) is:

• A. \( -97 \)
• B. \( -87 \)
• C. \( -77 \)
• D. \( -67 \)
• E. \( -57 \)

Hint:

If the sequence is decreasing, the common difference \( d \) will always be negative.

Answer 1

The Correct Option is C

Concept: The \( n^{th} \) term of an Arithmetic Progression (A.P.) is: \[ a_n = a + (n-1)d \] where \( a \) is the first term and \( d \) is the common difference.

Step 1: Identify A.P. parameters.
Given sequence: \[ 10,\ 7,\ 4,\ \dots \] First term: \[ a = 10 \] Common difference: \[ d = 7 - 10 = -3 \] Required term: \[ n = 30 \]

Step 2: Substitute into formula.
\[ a_{30} = 10 + (30 - 1)(-3) \] \[ = 10 + 29(-3) \]

Step 3: Simplify.
\[ = 10 - 87 \] \[ = -77 \]

Step 4: Final answer.
\[ \boxed{-77} \]