The sum of first 20 terms of the sequence 0.7,0.77, 0.777, …, is (A) (7)/(9)(99-10^-20) (B) (7)/

The sum of first 20 terms of the sequence 0.7,0.77, 0.777,...

Key Concepts

Geometric Series

A geometric series is a sum where each term is derived by multiplying the previous one by a constant ratio. This concept is crucial because it provides a systematic way to sum terms that increase or decrease exponentially. The formula for the sum of a finite geometric series, which involves the first term and the common ratio, is widely applicable in various mathematical problems involving repeated scaling.

Decimal Place Value

The decimal number system is based on place value, where a number is expressed as a sum of digits multiplied by powers of ten. Understanding this concept is essential for decomposing numbers into sums of fractions (e.g., tenths, hundredths, and so on), which is the basis for representing numbers with repeated digits as a series. This decomposition often allows such numbers to be reinterpreted as partial sums of a geometric series.

Series Summation Techniques

Techniques for summing series involve identifying the pattern in the sequence and applying appropriate formulas. For sequences that can be interpreted as geometric series or are constructed through iterative addition of terms with a common structure, using these summation techniques simplifies the process of finding the total of multiple terms. This concept is foundational in both elementary and advanced algebra.

Transcript

00:02 So in this question the sum of first 20 terms of the sequence 0 .7, 0 .77, 0 .77 etc.

00:16 So we want to find the sum of first 20 terms.

00:21 So let s n is the sum of n terms.

00:25 So s n is equal to 0 .7 plus 0 .7 plus point seven seven plus etc up to up to 20 terms up to 20 terms okay now which is equal to we can put seven outside the bracket so seven into point one plus point one plus point one 1 plus etc up to 20 terms now we can multiply and divided by 9 in both side that is multiply and divided by 9 so 7 by 9 and into 0 0 0 0 into 0 .9 plus 0 .99 plus 0 .99 plus 0 .9 plus etc up to 20 terms so which is equal to 7 by 9 into this point 9 can be written like 1 minus 0 .1 that is 1 minus 0 .1 is equal to 0 .9 here 1 minus 0 .01 1 minus 0 1 minus 0 1 1 0 1 plus 1 1 minus 0 .001 etc plus etc up to 20 terms okay now by adding all these ones 1 1 up to 20 terms there are total 20 ones are there so we can write this as 7 by 9 into 20 by 20 by 20 20 then minus rest of things point 1 which is equal to 0 .1 by 10 plus 1 by 100 or 1 by 10 square plus 1 by 10 rise to 3 plus etc up to 1 by 10 rise to 20 since there are 20 terms there are 20 terms so up to 1 by 10 by 10 x to 20 okay now here 1 by 10 here 1 by 10 plus 1 by 10 rise to 2 plus 1 by 10 rise to 3 plus etc up to 1 by 10 rise to 20 up to 10 rise to 20 is a g is a geometric progression gp with gp so here in this gp is equal to in this gp a is equal to 1 by 10 first term is 1 by 10 and it the common ratio is equal to common ratio r is equal to 1 by 10 1 by 10 1 by 10 is the common ratio in this gp we know that sum of 10 terms of a gp sum of n terms of a gp sum of n terms of a gp is equal to a into 1 minus r raised to n divided by 1 minus r r 1 minus r okay so by using this identity we can write sum of n terms in this in this in this ep is equal to 1 by 10 first term is 1 by 10 so 1 by 10 into 1 minus 1 minus 1 by 10 which is a nth term 1 10 rise to 20 divided by 1 minus 1 by 10 x 2...

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