Question : Find the $n$th term of the following sequence: 5 + 55 + 555 + ........+$T_n$
Option 1: $5(10^n-1)$
Option 2: $5^n(10^n-1)$
Option 3: $\frac{5}{9}(10^{n} - 1)$
Option 4: $(\frac{5}{9})^{n} (10^{n} -1)$
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Correct Answer: $\frac{5}{9}(10^{n} - 1)$
Solution :
Given: 5 + 55 + 555 + ........
$n$th term = 5 (1 + 11 + 111 + ... to $n$ terms)
Multiplying and divide by 9 in series,
$n$th term = $\frac{5}{9}$(9 + 99 + 999 +... to $n$ terms)
⇒ $n$th term = $\frac{5}{9}$[(10–1) + (10
2
–1) +...+ $(10^{n} -1)$]
⇒ $n$th term = $\frac{5}{9}(10^{n} –1)$
Hence, the correct answer is $\frac{5}{9}(10^{n} - 1)$.
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