If A be the first term and D
be common difference of an A.P.
Prove that S =" [2a +(n-1)d]
Answers
Answer:
Now nth term of the given Arithmetic Progression is a + (n - 1)d
Let the nth term of the given Arithmetic Progression = l
Therefore, a + (n - 1)d = l
Hence, the term preceding the last term is l – d.
The term preceding the term (l - d) is l - 2d and so on.
Therefore, S = a + (a + d) + (a + 2d) + (a + 3d) + …………………….. to n tems
Or, S = a + (a + d) + (a + 2d) + (a + 3d) + …………………….. + (l - 2d) + (l - d) + l ……………… (i)
Writing the above series in reverse order, we get
S = l + (l - d) + (l - 2d) + ……………. + (a + 2d) + (a + d) + a………………(ii)
Adding the corresponding terms of (i) and (ii), we get
2S = (a + l) + (a + l) + (a + l) + ……………………. to n terms
⇒ 2S = n(a + l)
⇒ S = n/2(a + l)
⇒ S = Numberofterms/2 (First term + Last term) …………(iii)
⇒ S = n/2[a + a + (n - 1)d], Since last term l = a + (n - 1)d
⇒ S = n/2[2a + (n - 1)d]
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Answer:
Suppose, S denote the sum of the first n terms of the Arithmetic Progression {a, a + d, a + 2d, a + 3d, a + 4d, a + 5d ……………...}.
Now nth term of the given Arithmetic Progression is a + (n - 1)d
Let the nth term of the given Arithmetic Progression = l
Therefore, a + (n - 1)d = l
Hence, the term preceding the last term is l – d.
The term preceding the term (l - d) is l - 2d and so on.
Therefore, S = a + (a + d) + (a + 2d) + (a + 3d) + …………………….. to n tems
Or, S = a + (a + d) + (a + 2d) + (a + 3d) + …………………….. + (l - 2d) + (l - d) + l ……………… (i)
Writing the above series in reverse order, we get
S = l + (l - d) + (l - 2d) + ……………. + (a + 2d) + (a + d) + a………………(ii)
Adding the corresponding terms of (i) and (ii), we get
2S = (a + l) + (a + l) + (a + l) + ……………………. to n terms
⇒ 2S = n(a + l)
⇒ S = n/2(a + l)
⇒ S = Numberofterms/2 (First term + Last term) …………(iii)
⇒ S = n/2[a + a + (n - 1)d], Since last term l = a + (n - 1)d
⇒ S = n/2[2a + (n - 1)d]