(iii) 2/5 + 3/5² + 2/5³ + 3/5⁴ + …. ∞

(iv) 10 – 9 + 8.1 – 7.29 + …. ∞

Solution:

(iii) 2/5 + 3/5² + 2/5³ + 3/5⁴ + …. ∞

We can write the given terms as,

(2/5 + 2/5³ + …) + (3/5² + 3/5⁴ + …)

where (a = 2/5, r = 1/25) for first part and (a = 3/25, r = 1/25) for second part

By making use of the formula,

$ {{S}_{\infty }}~=\text{ }a/\left( 1-r \right) $

$ =\left( \frac{2/5}{1-\frac{1}{25}} \right)+\left( \frac{3/5}{1-\frac{1}{25}} \right) $

$ =\left( \frac{10}{24}+\frac{3}{24} \right) $

$ =\frac{13}{24} $

(iv) 10 – 9 + 8.1 – 7.29 + …. ∞

We can write the above equation as:

S_∞ = 8 + 4√2 + 4 + …. ∞

Where we have a = 10, r = -9/10

By making use of the formula,

$ {{S}_{\infty }}~=\text{ }a/\left( 1-r \right) $

$ =\text{ }10\text{ }/\text{ }\left( 1-\left( -9/10 \right) \right) $

$ =\text{ }10\text{ }/\text{ }\left( 1\text{ }+\text{ }9/10 \right) $

$ =\text{ }10\text{ }/\text{ }\left( \left( 10+9 \right)/10 \right) $

$ =\text{ }10\text{ }/\text{ }\left( 19/10 \right) $

$ =\text{ }100/19 $

$ =\text{ }5.263 $