Question

solve the solution 1+4+7+10.............+x = 287

Answer 1

This is arithmetic series with
a = 1, d = 3

Sn = n/2 {2a + (n-1)d} = 287
n/2(2x1 + 3n - 3) = 287
n/2(3n - 1) = 287
3/2n² - n/2 - 287 = 0
3n² - n - 574 = 0    This is a quadratic equation with:
a = 3, b = -1, c = -574
n = 1+/- √1² - 4x3x-574
     -----------------------------
               2x3
n = 1 +/- 83
       ----------
           6

n = 84/6 = 14

'x' is the 14th tern

nth term =  a + (n-1)d
14th term = 1 + (14-1)3
                  =  40

∴ x = 40

Answer 2

$\Large{\textbf{\underline{\underline{According\;to\;the\;Question}}}}$

Given,

First term = 1 (a)

Common difference (d) = 4 - 1 = 3

Here,

Number of terms = n

S(n) = n

Using formula we have,

S(n) = n/2{2a + (n - 1)d}

287 = n/2{2 × 1 + (n - 1)(3)}

287 = n/2{2 + (n - 1)3}

574 = 3n² - n

3n² - n - 574 = 0

Splitting the middle term,

3n² - 42n + 41n - 574 = 0

3n(n - 14) + 41(n - 14) = 0

n = 14, -41/3 (Negative value is not acceptable)

n = 14

As we know that,

a + (n - 1)d = p

1 + (14 - 1)(3) = 3

1 + 13(3) = 3

p = 40

Therefore,

Value of p = 40