Find the sum of the sequence 7, 77, 777, 7777, ... to $n$ terms.

Step 1: Express terms in a general form

The given sequence is $7, 77, 777, \dots$. We need to find the sum of the first $n$ terms, denoted as $S_n$. 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 $10 + 10^2 + \dots + 10^n$ is given by the formula $a(r^n - 1)/(r - 1)$, 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.

More from this chapter

• 2.Testthefollowingseriesforconvergence. Ineachcasestatewhethertheconvergenceisabsoluteor conditional: (a) ∞ n=1 (−1)n−1 √n (b) ∞ n=1 (−1)n−1n n2+1 (c) ∞ n=1 (−1
• Solve the following equations for x 11. 3²x+4 + 1 = 2 · 3^(x+2) 12. 5²x+1 = 6 · 5^x - 1 13. 2²x - 2^(x+3) = -2^4 Solve for x and y: 14. 9^x = 3^(y-2), 81^y = 3²
• Find the sum of an infinite geometric series: 1 + 1/3 + 1/9 + 1/27 +...
• Prove by mathematical induction that 1³ + 2³ + 3³... + n³ = [n(n+1)/2]².
• Find the natural number a for which Σ_k=1^(n) f(a+k)=16(2^(n)-1), where the function f satisfies f(x+y)=f(x). f(y) for all natural numbers x, y and further f(1)
• Determine the convergence of the series Σ(n=1 to ∞) (n!)²/(2n)! using the ratio test.
• Find the sum of first n terms and the sum of first 5 terms of the geometric series 1+(2)/(3)+(4)/(9)+
• (i) If a, b, c, d are four distinct positive quantities in A.P., then show that b c>a d (ii) If a, b, c, d are four distinct positive quantities in G.P., then s

All Sequences and Series solutions →