(i) a = 7, d =\(\frac{21}{2}\) - 7 = \(\frac{7}{2}\)
Last term, an = 84
an = a + (n – 1)d
84 = 7 + (n – 1)\(\frac{7}{2}\)
84 = \(\frac{14 + 7n - 7}{2}\)
84 x 2 = 7 + 7n
168 = 7 + 7n
7n = 168 – 7
7n = 161
n = 23
Hence, sum of given A.P. is \(\frac{2093}{2}\)
(ii)
a = 34, d = 32 – 34 = -2
Last term, an = 10
an = a + (n – 1)d
10 = 34 + (n – 1) (-2)
10 = 34 – 2n + 2
2n = 34 – 10 + 2
2n = 26
n = 13
Sum of n terms,
Hence, Sum of given A.P. is 286
(iii)
a = 25, d = 3
Last term, an = 100
An = a + (n -1) d
100 = 25 (n – 1)3
75 = 3n – 3
78 = 3n
n = 26
sum of n terms,
= 13 [50 + 25 x 3]
= 13 x 125 = 1625
Hence, Sum of given A.P. is 1625
(iv)
a = 18, d =\(\frac{31}{2}\) -18 = \(\frac{-5}{2}\)
Last term, an = \(\frac{-99}{2}\)
an = a+ (n – 1)d
\(\frac{-99}{2}\)= 18 + (n – 1)\((\frac{-5}{2})\)
-99 = 36 + 5 – 5n
-140 = -5n
n = 28
= 7 (36 – 99)
= 7 (-63) = -441