Q.Express the recurring decimal $0.\overline{7}$ as a fraction using the sum to infinity of a GP.

$0.7777\ldots$ can be split into the repeating pattern $0.7 + 0.07 + 0.007 + \cdots$, which is a GP with first term 0.7 and common ratio 0.1.

Given: the recurring decimal $0.\overline{7} = 0.7777\ldots$

Step 1 — Split into a GP: $0.7777\ldots = 0.7 + 0.07 + 0.007 + 0.0007 + \cdots$. Each term is one-tenth of the previous one, so this is a GP with $a = 0.7$ and $r = 0.1$.

Step 2 — Confirm $S_\infty$ exists: $|r| = 0.1 < 1$, so the infinite sum is defined and equals the original decimal exactly.

Step 3 — Apply the formula: $S_\infty = \dfrac{a}{1-r} = \dfrac{0.7}{1-0.1} = \dfrac{0.7}{0.9}$. …

Taking $a$ as $0.7777$ (the whole decimal) instead of $0.7$ (just the first repeating block) — the GP's first term is the value …