than 250. (4) [15] QUESTION 3 3.1 If \( \frac{16}{3} \) and \( \frac{81}{2} \) are the \( 2^{\text {nd }} \) and the \( 7^{\text {th }} \) terms respectively of a geometric sequence 3.1.1 Write the sum of the first \( n \) terms of the sequence in sigma notation. 3.1.2 Is the sequence convergent or not? Justify your answer. (2) 3.1.3 Calculate the sum of the first 5 terms. (3) Copyright reserved Please turnover

In a geometric sequence, each term can be expressed as the first term multiplied by a common ratio raised to the power of its position in the sequence (n). From the given terms, we can denote the first term as \( a \) and the common ratio as \( r \). So, the \( 2^{nd} \) term can be represented as \( ar = \frac{16}{3} \) and the \( 7^{th} \) term can be expressed as \( ar^6 = \frac{81}{2} \). From the formula for the sum of the first \( n \) terms, \( S_n = a \frac{1 - r^n}{1 - r} \), we express this in sigma notation as \( S_n = \sum_{k=0}^{n-1} ar^k \). To determine if the sequence converges, we check the common ratio \( r \). If \( |r| < 1 \), the series converges, and we can compute the limit as \( n \) approaches infinity. If \( |r| \geq 1 \), the series diverges. Calculating the sum of the first 5 terms involves finding the specific values of \( a \) and \( r \) from the equations \( ar = \frac{16}{3} \) and \( ar^6 = \frac{81}{2} \). Once we have \( a \) and \( r \), we can substitute those values into \( S_5 = a \frac{1 - r^5}{1 - r} \) to get the desired sum.