Solve the equation for x: 1 + 4 + 7 + 10 + …. + x = 287

Q) Solve the equation for x:

1 + 4 + 7 + 10 + …. + x = 287

Ans:

We can see that it is AP with first term a = 1 and common difference d = 3

Let x be the value of n^th term and we need to find value of this n^th term.

Let’s start from n^th term:

The value of n^th term = a + (n – 1) d

Hence, x = 1 + (n – 1) 3

x = 3n – 2….. (i)

$\therefore n = \frac{x + 2}{3}$ ………………….. (ii)

Next, we know that, sum of n terms S_n = $\frac{n}{2} [2a + (n - 1)d]$

$\therefore$ 287 = $\frac{n}{2}$ [2 x 1 + (n – 1) 3 ]

574 = n (3n-1)

By substituting value of n from equation (ii)

574 = $(\frac{x + 2}{3})$ $[3 (\frac{x + 2}{3}) -1]$

(574)(3) = (x + 2) [ (x + 2) – 1]

1722 = (x + 2) (x + 1)

x² + 3 x – 1720 = 0

(x + 43) (x – 40) = 0

Hence, x = – 43 and x = 40

Here, we reject x = – 43 due to negative value and accept x = 40

Therefore, value of x = 40

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