An AP is given k,2k/3,k/3,0....calculate the sum of the first 30 terms
clearly d = -k/3
S30 = 30/2 (2k + 29(-k/2)) = -375k/2
small typo in Steve's answer:
last line should be
S30 = 30/2 (2k + 29(-k/3))
= 15(-23k/3)
= -115k
To calculate the sum of the first 30 terms of an Arithmetic Progression (AP), we need to find the formula for the nth term, and then use the formula for the sum of an AP.
In this case, the AP is given as k, 2k/3, k/3, 0, ...
To find the formula for the nth term, we can observe the pattern in the given terms. Notice that each term is related to the previous term by multiplying by a common ratio of 2/3. This means that the common difference (d) of the AP is k * 2/3.
The formula for the nth term of an AP is given by:
an = a1 + (n-1) * d
Here, a1 represents the first term of the AP, n represents the term number, and d represents the common difference.
Applying this formula to our AP, we have a1 = k and d = k * 2/3. So, the formula for the nth term becomes:
an = k + (n-1) * (k * 2/3)
Now, to find the sum of the first 30 terms of the AP, we can use the formula for the sum of an AP:
Sn = n/2 * (a1 + an)
Where Sn represents the sum of the first n terms of the AP.
Plugging in the values, we have:
S30 = 30/2 * (k + (30-1) * (k * 2/3))
Simplifying further:
S30 = 15 * (k + 29k * 2/3)
S30 = 15 * (k + 58k/3)
S30 = 15 * (k(1 + 58/3))
S30 = 15 * (k(61/3))
S30 = 305k
So, the sum of the first 30 terms of the given AP is 305k.