- Sum of the First 'N' Terms of an Arithmetic Progression
Sum of first n terms of an `"AP": "S" =(n/2)[2a + (n- 1)d]` The sum of n terms is also equal to the formula where l is the last term.
Arithmetic Progression a, a + d, a + 2d, a + 3d, . . . . . . . . . . . . a +(n - 1)d
In this progression a is the first term and d is the common difference. Let’s write the sum of first n terms as Sn.
Sn = [a] + [a + d] + . . . + [a+(n-2)d] + [a+(n-1)d] ..........(eq1)
Reversing the terms and rewritting the expression again,
Sn = [a+(n-1)d] + [a+(n-2)d] + . . . + [a + d ] + [a] ........(eq2)
On adding eq1 and eq2
2Sn = [a+a+(n-1)d] + [a + d+a+(n-2)d]+ . . . + [a+(n-2)d+ a + d]+ [a+(n-1)d+a]
2Sn = [2a+(n-1)d] + [2a+(n-1)d] + . . . + [2a+(n-1)d] . . . n times.
∴2Sn = n [2a+(n-1)d]
∴Sn = n/2 [2a+(n-1)d]
Ex. Let’s find the sum of first 100 terms of A.P. 14, 16, 18, . . . .
Here a= 14, d = 2, n = 100
`Sn = n/2 [2a+(n-1)d]`
`S100= 100/2 [2× 14+ (100-1)2]`
`S100= 50 [28 + (99)2]`
`S100= 50 [28 + 198]`
`S100= 50 `
∴ Sum of first 100 terms of given A.P. is 11,300
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Series 1: playing of 4
Ramkali required Rs 2,500 after 12 weeks to send her daughter to school. She saved Rs 100 in the first week and increased her weekly saving by Rs 20 every week. Find whether she will be able to send her daughter to school after 12 weeks.
What value is generated in the above situation?