2. Write in sigma notation: a) \( 1+2+3+4+5+6 \) b) \( 1+4+9+16+25+36+49 \) (c) \( 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\ldots+\frac{1}{10} \) d) \( 1+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\ldots+\frac{20}{21} \) e) \( 1.2+2.3+3.4+\ldots+ \) to \( n \) terms. f) \( 3+3+3+\ldots+ \) to \( n \) terms. g) \( a+a+a+\ldots+ \) to \( n \) terms. (h) \( p^{2}+3 p^{4}+5 p^{6}+\ldots+ \) to \( n \) terms. i) \( x^{n}+x^{n}-1 y+x^{n-2} y^{2}+x^{n-3} y^{3}+\ldots+x y^{n-1}+y^{n} \) i) \( x+2 x^{2}+6 x^{3}+24 x^{4}+120 x^{5}+\ldots \) to \( n \) terms.

Sure! Below are the sigma (summation) notations for each of the given expressions: --- **a)** \( 1 + 2 + 3 + 4 + 5 + 6 \) \[ \sum_{k=1}^{6} k \] --- **b)** \( 1 + 4 + 9 + 16 + 25 + 36 + 49 \) \[ \sum_{k=1}^{7} k^{2} \] --- **c)** \( 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \ldots + \frac{1}{10} \) \[ \sum_{k=1}^{10} \frac{1}{k} \] --- **d)** \( 1 + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6} + \ldots + \frac{20}{21} \) \[ \sum_{k=1}^{20} \frac{k}{k+1} \] --- **e)** \( 1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + n \cdot (n+1) \) (to \( n \) terms) \[ \sum_{k=1}^{n} k(k+1) \] --- **f)** \( 3 + 3 + 3 + \ldots + 3 \) (to \( n \) terms) \[ \sum_{k=1}^{n} 3 \] --- **g)** \( a + a + a + \ldots + a \) (to \( n \) terms) \[ \sum_{k=1}^{n} a \] --- **h)** \( p^{2} + 3p^{4} + 5p^{6} + \ldots \) (to \( n \) terms) \[ \sum_{k=1}^{n} (2k - 1) p^{2k} \] --- **i)** 1. **First Expression:** \( x^{n} + x^{n-1}y + x^{n-2}y^{2} + x^{n-3}y^{3} + \ldots + xy^{n-1} + y^{n} \) \[ \sum_{k=0}^{n} x^{n - k} y^{k} \] 2. **Second Expression:** \( x + 2x^{2} + 6x^{3} + 24x^{4} + 120x^{5} + \ldots \) (to \( n \) terms) \[ \sum_{k=1}^{n} k! \, x^{k} \] --- Each sigma notation succinctly represents the corresponding sum, with the index of summation (\( k \)) running through the specified range and the general term matching the pattern of the original series.