Find the sum of the following geometric series :√2 + 1/√2 + 1/2√

1.	Find the sum of the following geometric series :√2 + 1/√2 + 1/2√2 + ..... to 8 terms;
Answer» Common Ratio = r = \(\cfrac{\frac{1}{\sqrt{2}}}{\sqrt{2}}\) = \(\frac{1}{2}\) ∴ Sum of GP for n terms = \(\cfrac{a(r^n - 1)}{r-1}\) ........ (1) ⇒ a = √2, r = \(\frac{1}{2}\), n = 8 ∴ Substituting the above values in (1) we get ⇒ \(\cfrac{\sqrt{2}\bigg(\frac{1}{2})^8 -1\bigg)}{\frac{1}{2} - 1}\) ⇒ \(\cfrac{-\sqrt{2} \times 255 \times 2}{-1 \times 256}\) ⇒ \(\cfrac{255 \sqrt{2}}{128}\)

Find the sum of the following geometric series :√2 + 1/√2 + 1/2√2 + ..... to 8 terms;

Answer»

Common Ratio = r = \(\cfrac{\frac{1}{\sqrt{2}}}{\sqrt{2}}\) = \(\frac{1}{2}\)

∴ Sum of GP for n terms = \(\cfrac{a(r^n - 1)}{r-1}\) ........ (1)

⇒ a = √2, r = \(\frac{1}{2}\), n = 8

∴ Substituting the above values in (1) we get

⇒ \(\cfrac{\sqrt{2}\bigg(\frac{1}{2})^8 -1\bigg)}{\frac{1}{2} - 1}\)

⇒ \(\cfrac{-\sqrt{2} \times 255 \times 2}{-1 \times 256}\)

⇒ \(\cfrac{255 \sqrt{2}}{128}\)