Please help me. Is there a short cut way to arrive at this answer?
Observe the pattern of letters ACCCEEEEEGGGGGGG...
This pattern continues to repeat beginning with the letter A.
What will be the 170th letter?
Thank you
To find the 170th letter in the given pattern, we can analyze the pattern and identify its characteristics.
Looking at the given pattern ACCCEEEEEGGGGGGG..., we can notice that the pattern starts with 1 A, followed by 3 C's, then 5 E's, and 7 G's, continuing in a increasing sequence.
To calculate the number of letters in each group, we notice that the count of letters in each group follows an arithmetic progression:
Group 1: 1 letter (A)
Group 2: 1 + 2 = 3 letters (CC)
Group 3: 3 + 2 = 5 letters (EEE)
Group 4: 5 + 2 = 7 letters (GGGGGGG)
To find the length of each group, we use the formula for the sum of an arithmetic series:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the series, a is the first term, n is the number of terms, and d is the common difference between terms.
Applying this formula, we find the length of each group as follows:
Group 1: n = 1, a = 1, d = 2
S1 = (1/2)(2(1) + (1-1)2) = (1/2)(2 + 0) = 1
Group 2: n = 2, a = 1, d = 2
S2 = (2/2)(2(1) + (2-1)2) = (1)(2 + 2) = 4
Group 3: n = 3, a = 1, d = 2
S3 = (3/2)(2(1) + (3-1)2) = (3/2)(2 + 4) = 9
Group 4: n = 4, a = 1, d = 2
S4 = (4/2)(2(1) + (4-1)2) = (2)(2 + 6) = 16
From this analysis, we can deduce that the length of the nth group is given by the formula n^2.
Now, to find the group to which the 170th letter belongs, we can solve for n using the equation n^2 ≤ 170. In this case, n = 13.
Since each group has a length equal to its index squared, the 13th group will consist of 13^2 = 169 letters. Therefore, the 170th letter will be the first letter of the 14th group, which is H.
So, the answer to the question is H.