Question

in an ap: given an=56,d=7,sn=147 find a and​

Answer 1

$\red{ \qquad \underline{ \pmb{{ \mathbb{ \maltese  \:  COMPLETE \: \: QUESTION \:   \maltese }}}}}$

Q》In an AP, GIVEN

$\large \bf a_n = 56 \:,d = 7 \:,S_n = 147$

Then Find the value of a

and 'n'

$\Large\green{\qquad\underline{\pmb{{ \mathbb { \maltese  \:  GIVEN \:  \maltese }}}}}$

$\bf \: a_n(nth \: term) = 56 \\ \\ \bf d(common \:difference)= 7 \\ \\ \bf S_n(Sum \: of \: n \: terms)= 147$

$\purple{\qquad \qquad \underline{ \pmb{{ \mathbb{ \maltese  \:  REQUIRED \: \: INFO \:  \maltese }}}}}$

$\bigstar \mathfrak{ \text{S}um \: \: of \: \: n \: \: terms \: \: of \: \: \bf AP}\\ \bf S_n = \frac{n}{2} \{a + a_n \}$

$\bigstar \mathfrak{ \: \: nth \: term \: of \: an \: \bf AP} \\ \bf a_n = a + (n - 1)d$

$\Large \orange{\qquad \underline{ \pmb{{ \mathbb{ \maltese  \:SOLUTION \:  \maltese }}}}}$

$\large \mathfrak{ \text{W}e \: \text{K}now }$

$\bf S_n = \frac{n}{2} \{a + a_n \} \\ \\ \sf147 = \frac{n}{2} (a + 56) \\ \\ \bf294= n(a + 56)\:\:\:\:(i)$

$\bf a_n = a + (n - 1)d \\ \\ \sf56 = a + (n - 1)7 \\ \\ \sf56 = a + 7n - 7 \\ \\ \bf a = 63 -7n\:\:\:\:(ii)$

Putting (ii) in (i)

$\sf 294 = n(63 - 7n + 56) \\ \\ \sf 294 = n(119 - 7n) \\ \\ \sf294 = 119n - 7 {n}^{2} \\ \\ \sf 7 {n}^{2} - 119n + 294 = 0 \\ \\ \sf {n}^{2} - 17n + 42= 0 \\ \\ \sf{n}^{2} - 14n - 3n + 42 = 0 \\ \\ \sf n(n - 14) - 3(n - 14) = 0 \\ \\ \sf (n - 14)(n - 3) = 0 \\ \\ \implies\bf\color{magenta} \Large \boxed{\boxed{\{n = 14 \: \: or \: \: n = 3 \}}}$

When n = 14

$\sf a = 63 - 7(14) \\ \\ \implies \ \large\bf \{ a = - 35 \}$

When n = 3

$\sf a = 63 - 7(3) \\ \\ \implies \large\bf \{a = 42 \}$

$\huge\mathfrak{\text{S}o}\\\\\color{magenta} \boxed{\boxed{\Large\bf \{a = 42 \:\: or\:\:a=-35\}}}$

Answer 2

$S_n = \frac{n}{2} \{a + a_n \} \\ \\ 147 = \frac{n}{2} (a + 56) \\ \\ n(a + 56)=294 ...(Eq^n1)$

$a_n = a + (n - 1)d \\ \\ 56 = a + (n - 1)7 \\ \\ 56 = a + 7n - 7 \\ \\ a = 63 -7n...(Eq^n2)$

Put eq 2 in eq 1

We get

$n(63 - 7n + 56) =294\\ \\ n(119 - 7n) =294\\ \\ 119n - 7 {n}^{2} =294 \\ \\ \sf 7 {n}^{2} - 119n + 294 = 0 \\ \\ {n}^{2} - 17n + 42= 0 \\ \\ {n}^{2} - 14n - 3n + 42 = 0 \\ \\ (n - 14)(n - 3) = 0 \\ \\ \bf\implies[n = 14 \: \: or \: \: n = 3 ]$

If n = 14

$a = 63 - 7(14) \\ \\ \implies \ \large\bf a = - 35$

If n = 3

$a = 63 - 7(3) \\ \\ \implies \large\bf a = 42$