Q2. The sum of the first 16 terms of the sequence : 2,5,3,11,14,...,is

Answer 1

To find the sum of the first 16 terms of the sequence, we need to find a general formula for the nth term of the sequence.

Looking at the sequence, we can observe that the pattern alternates between adding and subtracting a certain value. The first term (n=1) is 2, the second term (n=2) is 5, and the third term (n=3) is 3. It appears that the pattern is adding a value of 3, then adding a value of 2, then subtracting a value of 4, and so on.

Using this pattern, we can write a formula for the nth term of the sequence:
an = an-1 + (-1)^(n+1) * (n+1), where a1 = 2

Now, let's find the sum of the first 16 terms of the sequence:
S16 = a1 + a2 + a3 + ... + a16

To find this sum, we substitute the formula for each term:
S16 = 2 + (2 + 3) + (2 + 3 - 4) + (2 + 3 - 4 + 4) + ... + (2 + 3 - 4 + 4 - 5 + ... + 16)

We can simplify this sum by canceling out terms:
S16 = 2 + 5 + 1 + 5 + 2 + 5 + 3 + 5 + ... + 16

Counting the number of terms, we can see that there are 8 sets of (5 + 2 + 5) and an additional 16 at the end:
S16 = 2 + 8(5 + 2 + 5) + 16

Simplifying the expression:
S16 = 2 + 8(12) + 16
S16 = 2 + 96 + 16
S16 = 114 + 16
S16 = 130

Therefore, the sum of the first 16 terms of the sequence is 130.