Question

solve the given equation (I) 1+4+7+10+.............+x=287​

Answer 1

$Given \: \: Question \: \: Is \: \\ \\ 1 + 4 + 7 + 10 + ... + x = 287 \\ \\ Answer \: \\ \\ this \: series \: is \: in \: ap \: as \: the \: common \: \\ difference \: is \: same. \\ \\ its \: sum \: is \: given \: \: by \: \\ \\ sn = \frac{n}{2} (a + an) \\ where \: sn \: is \: sum \: of \: n \: terms \\ n \: is \: number \: of \: terms \\ a \: is \: ist \: term \\ and \\ an \: is \: nth \: term \\ \\ n \: can \: be \: calculated \: as \\ \\ an = a + (n - 1)d \\ \\ here \: an = x \: \: \: a \: = 1 \\ d = 4 - 1 = 3 \\ \\ x = 1 + 3(n - 1) \\ \\ x = 1 + 3n - 3 \\ \\ x = 3n - 2 \\ \\ \frac{x + 2}{3} = n \\ \\ 287 = \frac{x + 2}{6} (x + 1) \\ \\ 287 \times 6 = (x + 2)(x + 1) \\ \\ x {}^{2} + 3x + 2 = 1722 \\ \\ x {}^{2} + 3x - 1720 = 0 \\ \\ x {}^{2} + 43x - 40x - 1720 = 0 \\ \\ x(x + 43) - 40(x + 43) = 0 \\ \\ (x - 40) = 0 \: \: \: \: \: or \: \: \: \: \: ( x+ 43) = 0 \\ \\ x = 40 \: \: \: or \: \: \: \: x = - 43$

Answer 2

Question:

→ $\\ 1 + 4 + 7 + 10 + ... + x = 287$

Answer:

This series is in AP as the common difference is same.

it's sum is given by

$\begin{lgathered} \\ \\ sn = \frac{n}{2} (a + an) \\ Where \: sn \: is \: Sum \: Of \: n \: Terms \\ n \: Is \: Number \: Of \: Terms \\ a \: is \: ist \: term \\ and \\ an \: is \: nth \: term \\ \\ n \: can \: be \: calculated \: as \\ \\ an = a + (n - 1)d \\ \\ here \\ \: an = x \: \: \: a \: = 1 \\ d = 4 - 1 = 3 \\ \\ x = 1 + 3(n - 1) \\ \\ x = 1 + 3n - 3 \\ \\ x = 3n - 2 \\ \\ \frac{x + 2}{3} = n \\ \\ 287 = \frac{x + 2}{6} (x + 1) \\ \\ 287 \times 6 = (x + 2)(x + 1) \\ \\ x {}^{2} + 3x + 2 = 1722 \\ \\ x {}^{2} + 3x - 1720 = 0 \\ \\ x {}^{2} + 43x - 40x - 1720 = 0 \\ \\ x(x + 43) - 40(x + 43) = 0 \\ \\ (x - 40) = 0 \: \: \: \: \: or \: \: \: \: \: ( x+ 43) = 0 \\ \\ x = 40 \: \: \: or \: \: \: \: x = - 43\end{lgathered}$