Find the sum of the sequence 7, 77, 777, 7777, ... to n terms.
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Step-by-Step Solution
Step 1: Express terms in a general form
The given sequence is 7,77,777,…. We need to find the sum of the first n terms, denoted as Sn. Each term consists of the digit 7 repeated a certain number of times.
Step 2: Factor out 7 and rewrite terms
We can factor out 7 from each term in the sum. This simplifies the expression and allows us to work with a sequence of ones.
Step 3: Convert terms to powers of 10
To express the terms in a more manageable form, we multiply and divide by 9. This allows us to write each term as a difference of powers of 10, for example, 9=10−1, 99=100−1, and so on.
Step 4: Separate into two geometric progressions
Now we can separate the sum into two parts: a sum of powers of 10 and a sum of n ones. The sum of powers of 10 forms a geometric progression.
Step 5: Apply geometric series formula
The sum of the geometric progression 10+102+⋯+10n is given by the formula a(rn−1)/(r−1), where a=10 and r=10. The sum of n ones is simply n.
Step 6: Simplify the expression
Finally, we simplify the expression by performing the subtraction in the denominator and distributing the terms.