Question

Sum of first 20 terms: 5, 11, 19, 29, 41

Answer 1

The given sequence is: 5, 11, 19, 29, 41, ...
We observe that this is an arithmetic progression (AP) with the first term \( a = 5 \) and the common difference \( d = 6 \) (since 11 - 5 = 6, 19 - 11 = 8, etc.).

The formula for the sum of the first \( n \) terms of an arithmetic progression is:
\[ S_n = \frac{n}{2} \cdot \left[ 2a + (n-1) \cdot d \right] \] Where: - \( a \) is the first term, - \( d \) is the common difference, - \( n \) is the number of terms. 
Step 1: Substitute the known values into the formula:
Given: \( a = 5 \), \( d = 6 \), and \( n = 20 \), we can substitute these values into the formula: \[ S_{20} = \frac{20}{2} \cdot \left[ 2 \cdot 5 + (20 - 1) \cdot 6 \right] \] \[ S_{20} = 10 \cdot \left[ 10 + 19 \cdot 6 \right] \] \[ S_{20} = 10 \cdot \left[ 10 + 114 \right] \] \[ S_{20} = 10 \cdot 124 = 1240 \] Final Answer:
The sum of the first 20 terms is 1240.