NCERT Class XI Mathematics - Sequences and Series - Solutions
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Question : 90
Total: 106
The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.
Solution:
Let a be the first term, d be the common difference and n be the last term. Then, the first 4 terms will be a, a + d, a + 2d, a + 3d and the last 4 terms will be n, n – d, n – 2d, n – 3d.
According to question, a + (a + d) + (a + 2d) + (a + 3d) = 56
⇒ 4a + 6d = 56 ⇒ 4(11) + 6d = 56 (Since a = 11)
⇒ 6d = 56 – 44 ⇒ 6d = 12 ⇒ d = 2
and n + (n – d) + (n – 2d) + (n – 3d) = 112
⇒ 4n – 6d = 112 ⇒ 4n – 6(2) = 112 (Since d = 2)
⇒ 4n = 112 + 12 ⇒ 4n = 124 ⇒ n = 31
Let n be the mth term of the A.P., then
n = a + (m – 1)d ⇒ 31 = 11 + (m – 1) 2
⇒ (m – 1) 2 = 20 ⇒ m – 1 = 10 ⇒ m = 11
∴ number of terms are 11.
According to question, a + (a + d) + (a + 2d) + (a + 3d) = 56
⇒ 4a + 6d = 56 ⇒ 4(11) + 6d = 56 (Since a = 11)
⇒ 6d = 56 – 44 ⇒ 6d = 12 ⇒ d = 2
and n + (n – d) + (n – 2d) + (n – 3d) = 112
⇒ 4n – 6d = 112 ⇒ 4n – 6(2) = 112 (Since d = 2)
⇒ 4n = 112 + 12 ⇒ 4n = 124 ⇒ n = 31
Let n be the mth term of the A.P., then
n = a + (m – 1)d ⇒ 31 = 11 + (m – 1) 2
⇒ (m – 1) 2 = 20 ⇒ m – 1 = 10 ⇒ m = 11
∴ number of terms are 11.
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