Find the sum of the arithmetic series \( 4+9+14+\ldots+154 \) Find the sum of the arithmetic series \( -7-5-3-\ldots-25 \) Determine \( n \) in each of the arithmetic seriec:

Let's solve each part of your problem step by step. --- ### **1. Sum of the Arithmetic Series \( 4 + 9 + 14 + \ldots + 154 \)** **Given:** - **First term (\(a_1\))**: 4 - **Last term (\(a_n\))**: 154 - **Common difference (\(d\))**: \(9 - 4 = 5\) **a. Finding the Number of Terms (\(n\))** Use the formula for the \(n\)-th term of an arithmetic series: \[ a_n = a_1 + (n - 1)d \] Plugging in the known values: \[ 154 = 4 + (n - 1) \times 5 \] \[ 150 = (n - 1) \times 5 \] \[ n - 1 = \frac{150}{5} = 30 \] \[ n = 31 \] **b. Calculating the Sum (\(S_n\))** Use the sum formula for an arithmetic series: \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] \[ S_{31} = \frac{31}{2} \times (4 + 154) = \frac{31}{2} \times 158 = 31 \times 79 = 2,\!449 \] --- ### **2. Sum of the Arithmetic Series \( -7 -5 -3 -\ldots-25 \)** **Given:** - **First term (\(a_1\))**: \(-7\) - **Last term (\(a_n\))**: \(-25\) - **Common difference (\(d\))**: To determine **Determining the Common Difference (\(d\))** Observing the series: \[ -7,\, -5,\, -3,\, \ldots,\, -25 \] Each term increases by 2: \[ -5 - (-7) = 2,\quad -3 - (-5) = 2,\quad \text{etc.} \] Thus, \(d = 2\). **However**, there's an inconsistency. Starting at \(-7\) and increasing by \(2\) will lead to larger (less negative) numbers, not smaller. To reach \(-25\), the common difference should be negative. Assuming this was intended, we'll use \(d = -2\). **a. Finding the Number of Terms (\(n\))** Using the corrected common difference: \[ a_n = a_1 + (n - 1)d \] \[ -25 = -7 + (n - 1)(-2) \] \[ -25 + 7 = -2(n - 1) \] \[ -18 = -2(n - 1) \] \[ n - 1 = \frac{-18}{-2} = 9 \] \[ n = 10 \] **b. Calculating the Sum (\(S_n\))** \[ S_n = \frac{n}{2} \times (a_1 + a_n) \] \[ S_{10} = \frac{10}{2} \times (-7 + (-25)) = 5 \times (-32) = -160 \] --- ### **Summary of Results** 1. **First Series (\(4 + 9 + 14 + \ldots + 154\))** - **Number of Terms (\(n\))**: 31 - **Sum (\(S_n\))**: 2,\!449 2. **Second Series (\(-7 -5 -3 -\ldots-25\))** - **Number of Terms (\(n\))**: 10 - **Sum (\(S_n\))**: \(-160\) --- **Note:** It's essential to ensure the common difference aligns with the progression of the series to avoid inconsistencies.