Sigma Notation

[Maximum mark: 6]

Consider the sum $\displaystyle S = \sum_{k = 4}^l (2k-3)$, where $l$ is a positive integer greater than $4$.

1. Write down the first three terms of the series. [2]
   

2. Write down the number of terms in the series. [1]
   

3. Given that $S = 725$, find the value of $l$. [3]
   

[Maximum mark: 6]

Let $u_n = 5n-1$, for $n \in \mathbb{Z}^+$.

1. 1. Using sigma notation, write down an expression for $u_1 + u_2 + u_3 + \dots + u_{10}$.

   2. Find the value of the sum from part (a) (i). [4]
      

A geometric sequence is defined by $v_n = 5\times 2^{n-1}$, for $n \in \mathbb{Z}^+$.

2. Find the value of the sum of the geometric series $\medmath{\displaystyle\sum_{k = 1}^6}\hspace{0.1em}v_k$.[2]
   

[Maximum mark: 6]

Let $u_n = 4n+1$, for $n \in \mathbb{Z}^+$.

1. 1. Using sigma notation, write down an expression for $u_1 + u_2 + u_3 + \dots + u_{20}$.

   2. Find the value of the sum from part (a) (i). [4]
      

A geometric sequence is defined by $v_n = 9\times 4^{n-1}$, for $n \in \mathbb{Z}^+$.

2. Find the value of the sum of the geometric series $\medmath{\displaystyle\sum_{k = 1}^5}\hspace{0.1em}v_k$.[2]