To calculate the sum of the sequence: 7, 77, 777, 7777, to n Terms:

When trying to find the sum of a sequence like 7, 77, 777, 7777, and so on up to n terms, we realize that this sequence does not follow the typical rules of an arithmetic progression (AP) or a geometric progression (GP).

Unlike AP, the difference between consecutive terms is not constant, and unlike GP, the ratio is not fixed. However, we can relate it to a GP by rewriting the terms in a more convenient form.

Now that sum of the sequence can be expressed as:

S_n = 7 + 77 + 777 + …….. to n terms

By factoring out 7, we get:

S_n = 7(1 + 11 + 111 +…… to n terms)

Next, to simplify further, we divide and multiply by 9:

S_n = (7/9) (9 + 99 + 999 +……..to n terms)

This can be broken down into two geometric progressions (GPs). First, let’s focus on rewriting the sequence in a more structured way:

S_n = (7/9)[(10 – 1) + (100 – 1) + (1000 – 1) +……..to n terms]

Which simplifies to:

S_n = (7/9)[(10 + 100 + 1000 +……to n terms) – (1 + 1 + 1 +……. to n terms)]

Now, let’s analyze these two new sequences:

GP 1:

This is a geometric progression where:

The first term (a) is 10.

The common ratio (r) is 10.

Thus, the sum of the first n terms of this Geometric progression is given by:

S_GP1 = a(rⁿ – 1)/(r – 1)

Substituting a = 10 and r = 10 into the formula:

Thus, the sum is S_GP1 = 10(10ⁿ – 1)/(10 – 1) = 10(10ⁿ – 1)/9

GP 2:

This is a constant sequence where every term is 1. Therefore, the sum of this sequence is simply n:

S_GP2 = 1 x n = n

Final Formula for the Sum of the Sequence

Now that we have both sums, we can substitute them into our main equation:

S_n = (7/9)[{10(10^n – 1)}/9} – n]

This is the general formula for the sum of the sequence 7, 77, 777, 7777, and so on to n terms.

Finding the Sum of the First 10 Terms

To find the sum of the first 10 terms, we substitute n = 10 into the formula:

Thus By calculating this, you will find the sum of the first 10 terms as 8641975300.

Conclusion

In conclusion, though the sequence may initially seem complex, breaking it down into two geometric progressions provides an efficient method for finding its sum. This approach not only makes the calculation simpler but also reveals an interesting pattern in the structure of the sequence. Transitioning through these steps allows for a clearer understanding, making it easier to compute even larger sums.