Question

The A.P. \( 8, 10, 12, \ldots \) has 60 terms. Find the sum of the last 20 terms.

Hint:

To find the sum of a specific part of an A.P., first find the first and last terms of the segment, then use the sum formula for an A.P.

Answer 1

Step 1: Identify the given values.}
For an arithmetic progression (A.P.), the general form of the nth term is given by: \[ T_n = a + (n-1)d \] Where: - \( a \) is the first term. - \( d \) is the common difference. - \( n \) is the number of terms. From the given A.P., we have: - \( a = 8 \) - \( d = 2 \) - The total number of terms is 60. We need to find the sum of the last 20 terms.
Step 2: Find the 41st term.}
The 41st term is the first term of the last 20 terms. Using the formula for the nth term: \[ T_{41} = 8 + (41 - 1) \times 2 = 8 + 40 \times 2 = 8 + 80 = 88 \]
Step 3: Use the sum formula for an A.P.}
The sum of the last 20 terms is given by the formula: \[ S = \frac{n}{2} \times (T_1 + T_n) \] Where \( n \) is the number of terms (20), \( T_1 \) is the first term of the last 20 terms (88), and \( T_n \) is the last term (60th term).
Step 4: Find the 60th term.}
Using the nth term formula again: \[ T_{60} = 8 + (60 - 1) \times 2 = 8 + 59 \times 2 = 8 + 118 = 126 \]

Step 5: Calculate the sum of the last 20 terms.}
Now we calculate the sum: \[ S = \frac{20}{2} \times (88 + 126) = 10 \times 214 = 2140 \] % Final Answer Final Answer:
The sum of the last 20 terms is \( 2140 \).