Question 8 If the sum of n terms of an A.P. is (pn + qn2), where p and q are constants, find the common difference.
Class X1 - Maths -Sequences and Series Page 185
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Given,
S_n = pn + qn² , where P and q are constants.
we know, formula ,
T_n = S_n - S_{n-1}
where
S_n is sum of n terms , S_{n-1} is sum of (n-1)terms .
T_n = Pn + qn² - P(n-1) - q(n-1)²
= P(n - n + 1) + q(n² - n² + 2n -1)
= P + q(2n -1)
= P + q(2n -2 + 1)
= P + q(2n -2) + q
= (p +q) + 2q(n-1)
now,
compare it with T_n = a + (n-1)d
we observed that,
a = (p +q) and d = 2q
hence, common difference (d) = 2q
S_n = pn + qn² , where P and q are constants.
we know, formula ,
T_n = S_n - S_{n-1}
where
S_n is sum of n terms , S_{n-1} is sum of (n-1)terms .
T_n = Pn + qn² - P(n-1) - q(n-1)²
= P(n - n + 1) + q(n² - n² + 2n -1)
= P + q(2n -1)
= P + q(2n -2 + 1)
= P + q(2n -2) + q
= (p +q) + 2q(n-1)
now,
compare it with T_n = a + (n-1)d
we observed that,
a = (p +q) and d = 2q
hence, common difference (d) = 2q
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