the AS: \( 2+9+16+\ldots . \). (to 251 Write down the fouth term of th Calculate the \( 251^{\text {st }} \) term of the s Express the series in sigma notat

To solve the problem, we need to analyze the arithmetic series given by \( 2, 9, 16, \ldots \). ### Step 1: Identify the pattern The first term \( a_1 = 2 \) and the second term \( a_2 = 9 \). The difference between the second and first term is: \[ a_2 - a_1 = 9 - 2 = 7 \] The third term \( a_3 = 16 \) gives: \[ a_3 - a_2 = 16 - 9 = 7 \] This indicates that the series has a common difference \( d = 7 \). ### Step 2: Write the general term The \( n^{\text{th}} \) term of an arithmetic series can be expressed as: \[ a_n = a_1 + (n-1) \cdot d \] Substituting the known values: \[ a_n = 2 + (n-1) \cdot 7 \] This simplifies to: \[ a_n = 2 + 7n - 7 = 7n - 5 \] ### Step 3: Find the fourth term To find the fourth term \( a_4 \): \[ a_4 = 7 \cdot 4 - 5 = 28 - 5 = 23 \] ### Step 4: Calculate the \( 251^{\text{st}} \) term To find the \( 251^{\text{st}} \) term \( a_{251} \): \[ a_{251} = 7 \cdot 251 - 5 \] Now, let's calculate \( a_{251} \): \[ a_{251} = 1757 - 5 = 1752 \] ### Step 5: Express the series in sigma notation The series can be expressed in sigma notation as: \[ \sum_{n=1}^{251} (7n - 5) \] ### Summary of Results - The fourth term of the series is \( 23 \). - The \( 251^{\text{st}} \) term of the series is \( 1752 \). - The series in sigma notation is \( \sum_{n=1}^{251} (7n - 5) \).