\[ \sum_{j=1}^{6} 6 j \] How many terms are in the series?

The given series is: \[ \sum_{j=1}^{6} 6j \] This notation represents the sum of terms where the variable \( j \) takes on integer values starting from 1 up to 6. Let's break it down: - **Lower limit**: \( j = 1 \) - **Upper limit**: \( j = 6 \) - **Term**: \( 6j \) **Calculating the Number of Terms:** To determine how many terms are in the series, subtract the lower limit from the upper limit and add 1 (since both endpoints are included): \[ \text{Number of terms} = (6 - 1) + 1 = 6 \] **Listing the Terms:** For clarity, let's list out each term in the series by substituting \( j \) with values from 1 to 6: 1. When \( j = 1 \): \( 6 \times 1 = 6 \) 2. When \( j = 2 \): \( 6 \times 2 = 12 \) 3. When \( j = 3 \): \( 6 \times 3 = 18 \) 4. When \( j = 4 \): \( 6 \times 4 = 24 \) 5. When \( j = 5 \): \( 6 \times 5 = 30 \) 6. When \( j = 6 \): \( 6 \times 6 = 36 \) So, the series expands to: \[ 6 + 12 + 18 + 24 + 30 + 36 \] **Conclusion:** There are **6 terms** in the series.