the 19th and 22nd term of an AP are 29 and 55 respectively Find the sum of it first sixty term
a+18d = 29
a+21d = 55
so, 3d = 26 and a = -127
S60 = 60/2 (2(-127)+59*26/3) = 7720
To find the sum of the first sixty terms of an arithmetic progression (AP), we need to determine the common difference (d) and use the formula for the sum of an AP.
Given that the 19th term (a_{19}) is 29 and the 22nd term (a_{22}) is 55, we can find the common difference using these two terms.
Step 1: Find the common difference (d)
We know that a_{22} = a_{19} + 3d, where "3d" represents three times the common difference, as there are three terms between the 19th and 22nd term.
Substitute the given values:
55 = 29 + 3d
Simplify the equation:
3d = 26
Divide both sides by 3:
d = 8.67 (approximately)
Step 2: Find the first term (a_{1})
To find the first term (a_{1}), we need to subtract three times the common difference from the 19th term.
a_{1} = a_{19} - 3d
Substitute the given values:
a_{1} = 29 - 3(8.67)
a_{1} = 29 - 26
a_{1} = 3
Step 3: Find the sum of the first sixty terms (S_{60})
The formula to find the sum of the first n terms of an AP is:
S_{n} = (n/2)[2a_{1} + (n-1)d]
Substitute the given values:
S_{60} = (60/2)[2(3) + (60-1)(8.67)]
S_{60} = 30[6 + 59(8.67)]
S_{60} = 30[6 + 511.53]
S_{60} = 30(517.53)
S_{60} = 15,525.9 (approximately)
Therefore, the sum of the first sixty terms of the given arithmetic progression is approximately 15,525.9.