Question

Question no. 1 . Pls give detail step

Answer 1

We know that the n'th term of an AP is,

a_n = a + (n - 1) d

a_n = a + dn - d

a_n = dn + (a - d)

This equation says that the n'th term is a linear equation in x where coefficient of n is the common difference and the sum of this coefficient and the constant is the first term.

Thus, in the equation a_n = 4n - 3,

d = 4 ; a = 4 - 3 = 1

We know, the sum of n terms of the AP,

S_n = n (2a + (n - 1)d) / 2

S_n = n (dn + 2a - d) / 2

S_n = (d / 2) n² + (a - (d / 2)) n

This equation shows that the sum is a quadratic equation in n where coefficient of n² is half the common difference and the sum of coefficients of n² and n is the first term. Thus,

d / 2 = 2 ; a - (d / 2) = 1 - 2 = - 1

Thus the sum is,

S_n = 2n² - n

S_n = n(2n - 1)

Hence, for n = 10, sum of first 10 terms is,

S_n = 10 (2 × 10 - 1)

S_n = 190

Hence (a) is the answer.

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