Question

3) Write concisely, using sigma notation:

(f) 1 + 1/3 + ( 1/3)² + ( 1/3)³ + · · · (to 8 terms)

(g) 2/5 − (2/5)² + ( 2/5)³ − · · · (to 20 terms)

(h) 1/1 + 2/3 + 3/5 + 4/7 + · · · (to n terms)

(i) 1/1 − 3/4 + 5/7 − 7/10 + · · · (to n terms)

(j) 1(1) + 2(5) + 3(5²) + 4(5³) + 5(5⁴) + · · · (to n terms)

(k) 1(2)(3) − 3(5)(3²) + 5(8)(3³) − 7(11)(3⁴) + · · · (to n terms).

Answer 1

(f)

$$1 + \dfrac{1}{3} + ( \dfrac{1}{3})^2 + ( \dfrac{1}{3})^3 + · · ·+( \dfrac{1}{3} )^7=\displaystyle\sum_{n=1}^8(\dfrac{1}{3})^{n-1}$$

(g)

$$\dfrac{2}{5} - (\dfrac{2}{5})^2 + ( \dfrac{2}{5})^2 - ( \dfrac{2}{5})^3 + · · ·-(\dfrac{2}{5})^{20}=\displaystyle\sum_{n=1}^{20}(-1)^{n+1}(\dfrac{2}{5})^{n}$$

(h)

$$\dfrac{1}{1} + \dfrac{2}{3} + \dfrac{3}{5} + \dfrac{4}{7} + · · ·+ \dfrac{n}{2n-1}=\displaystyle\sum_{i=1}^n\dfrac{i}{2i-1}$$

(i)

$$\dfrac{1}{1} - \dfrac{3}{4} + \dfrac{5}{7} - \dfrac{7}{10} + · · ·+ (-1)^{n+1}\dfrac{2n-1}{3n-2}$$

$$=\displaystyle\sum_{i=1}^n(-1)^{i+1}\dfrac{2i-1}{3i-2}$$

(j)

$$1(1)+2(5)+3(5^2)+4(5)^3+5(5)^4+...+n(5^{n-1})$$ $$=\displaystyle\sum_{i=1}^{n}n(5^{n-1})$$

(k)

$$1(2)(3)-3(5)(3^2)+5(8)(3^3)-7(11)(3^4)+...$$

$$+(-1)^{n+1}(2n-1)(3n-1)(3^n)$$

$$=\displaystyle\sum_{i=1}^n(-1)^{i+1}(2i-1)(3i-1)(3^i)$$