If $a_{n}$ is a sequence such that $a_{1} = 1$ , $a_{2} = 1 + a_{1}$ , $a_{3} = 1 + a_{1} a_{2}$ , $a_{n + 1} = 1 + a_{1} a_{2} a_{3} . . . . . . a_{n}$ then find the value of $\frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + \frac{1}{a_{4}} . . . . . \infty$

Open in App

Solution

$a_{1} = 1, a_{2} = 1 + a_{1} = 2 a_{3} = 1 + a_{1} a_{2} = 3, a_{4} = 1 + a_{1} a_{2} a_{3} = 7 a_{n + 1} = 1 + a_{1} a_{2} . . . . a_{n} = 1 + (a_{n} - 1) a_{n} (a_{n} = 1 + a_{1} a_{2} . . . . a_{n - 1}) a_{n + 1} = a_{n}^{2} - a_{n} + 1 a_{n + 1} - 1 = a_{n}^{2} - a_{n} \frac{1}{a_{n + 1} - 1} = \frac{1}{a_{n}^{2} - a_{n}} \frac{1}{a_{n + 1} - 1} = \frac{1}{a_{n} - 1} - \frac{1}{a_{n}} \frac{1}{a_{n}} = \frac{1}{a_{n} - 1} - \frac{1}{a_{n + 1} - 1} \frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + \frac{1}{a_{4}} + . . . . + \frac{1}{a_{n}} + . . \infty = \frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + (\frac{1}{a_{4} - 1} - \frac{1}{a_{5} - 1}) + (\frac{1}{a_{5} - 1} - \frac{1}{a_{6} - 1}) + . . . + (\frac{1}{a_{n} - 1} - \frac{1}{a_{n + 1} - 1}) = \frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + \frac{1}{a_{4} - 1} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{6} = 2 \frac{1}{a_{1}} + \frac{1}{a_{2}} + \frac{1}{a_{3}} + . . . . + \frac{1}{a_{n}} + . . . \infty = 2$

Suggest Corrections

Similar questions

a_{1}, a_{2},

a_{n}

a_{1} = 0

\frac{a_{3}}{a_{2}} + \frac{a_{4}}{a_{3}} + \frac{a_{5}}{a_{4}} +

+ \frac{a_{n}}{a_{n - 1}} - a_{2} (\frac{1}{a_{2}} + \frac{1}{a_{3}} + . . . + \frac{1}{a_{n - 2}}) = \frac{a_{n - 1}}{a_{2}} + \frac{a_{2}}{a_{n - 1}}

a_{1}, a_{2}, a_{3}, . . ., a_{n}

\frac{1}{a_{1} a_{2}} + \frac{1}{a_{2} a_{3}} + \frac{1}{a_{3} a_{4}} + . . . \frac{1}{a_{n - 1} a_{n}} = \frac{n - 1}{a_{1} a_{n}}

\frac{a_{1}}{a_{1}} + \frac{a_{2}}{a_{2}} + \frac{a_{3}}{a_{3}} + \dots \frac{a_{n}}{a_{n}} = \frac{1}{1} + \frac{1}{a_{1}} + \frac{a_{1}}{a_{2}} + \frac{a_{2}}{a_{3}} + \dots \frac{a_{n - 2}}{a_{n - 1}}

A_{1} A_{2} A_{3} . . . A_{n}

n

\frac{1}{A_{1} A_{2}} = \frac{1}{A_{1} A_{3}} + \frac{1}{A_{1} A_{4}},

a_{1}, a_{2}, a_{3} . . . . a_{n} \in R^{+}

a_{1} . a_{2} . a_{3} . . . . a_{n} = 1

(1 + a_{1} + a_{1}^{2}) (1 + a_{2} + a_{2}^{2}) (1 + a_{3} + a_{3}^{2}) . . . . (1 + a_{n} + a_{n}^{2})

Join BYJU'S Learning Program