How do I explain this pattern mathematically?

Question

I know what the pattern is doing, but how do I articulate it in mathematical terms? Also, I don't understand part b. How can this pattern not work? Doesn't it just continue?

a.) What is the pattern?
b.) How many addends are there the first time the pattern no longer works?

1 + 11 =
1 + 11 + 111 =
1 + 11 + 111 + 1111 =

(The sums are 12, 123, and 1234 respectively, if that helps.)

" isn't is already expressed mathematically?"

Haha, yes you are correct. I meant to say verbally; sorry, it's been a long day.

Answer 1

Each one is itself a geometric series with
a = 1, r = 10, n = ?
Let Sn = 1111..
eg S1 = 1, S2 = 11, S4 = 1111 etc
Sn = 1 + 10 + 100 + 1000 + ...
= a*(1-r^n)/(1-r)
= (1 - 10^n) / (-9)
= (10^n - 1) / 9
This is the value of each string.

Then each of those lines, Ln
= ∑ Sn
= ∑ (10^n - 1) / 9

Answer 2

you can use a recursive function

let K sub n be the first term = Σ10^n + Ksub(n-1)=1+0 because K sub -1 is 0

then any subsequent term = k sub(n-1)+ 10^n
then take their sum Σ └prev term┘
............................infinity
your series is = Σ 10^n + Ksub(n-1)
...........................n=0.......

B. the first nine terms will give you Σ=123456789 ( the same number of times the terms were added) at the tenth term the last digit doesn't match the number of terms. ie the 5ht term ends in 5, the 6th term ends in 6 etc....the 9th term will end in a 9 but the 10 ends in a 0 and so will the 20th, 30th etc .. a new pattern emerges then.... Try it out from the 11th term on to say 25 and you will see what I am saying.

Dr.D's answer is partially correct because it only gives you a the nth term by itself. if you want the sum total of the previous n-1 terms added to your nth term then:

= Σ(10^n - 1) / 9 is a more correct way of expressing your idea with a small n=1 under the sigma and an infinity sign above sigma.

OK I see he corrected his answer!

Answer 3

isn't is already expressed mathematically?