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Question 55
Let $$a_1, a_2, ......... a_{19}$$ be the first 19 terms of an arithmetic progression where $$a_1 + a_8 + a_{12} + a_{19} = 224$$. The sum $$a_1 + a_2 + a_3 + ........ + a_{19}$$ is equal to
Solution
Let first term $$a_1 = a$$ and common difference = d
$$a_2 = a + d; a_3 = a + 2d$$
Now,
$$a_1 + a_8 + a_{12} + a_{19} = 224$$
a + a + 7d + a + 11d + a + 18d = 224
4a + 36d = 224
a + 9d = 56 ....... (1)
= $$a_1 + a_2 + a_3 + ........ + a_{19}$$
= a + a + d + a + 2d + .......... + a + 18d
= 19a + 171d
= 19(a + 9d)
= 19 * 56
= 1064
