9 term of an Arithmetic Progression is 19.The sum of 4th and 7th term

1.	9 term of an Arithmetic Progression is 19.The sum of 4th and 7th term is 24. Find theArithmetic Progression.
Answer» $\underbrace{\underline{\bf{\bigstar\:Required\:Answer:-}}}$ $\\$ ➡ Given : $\\$ 9th Term of the A.P. = 19 4th Term + 7th Term = 24 $\\$ ➡ To Find : $\\$ The Arithmetic Progression (AP). $\\$ ➡ SOLUTION : $\\$ We know that the $\sf n^{th}$ term of an AP is GENERALISED as - $\leadsto \: \boxed{ \sf a _{n} =a + (n - 1)d }$ Here, a = FIRST term of the AP d = common difference n = no. of terms an = n th term $\\$ $\textit{\textbf{\dag\:According\:to\:the\:Question:}}$ $\\$ $\colon \longrightarrow \sf \: 9 {}^{th} \: term = 19 \\ \\$ $\colon \longrightarrow \sf \: a + 8d = 19 \\ \\$ $\colon \longrightarrow \: \boxed{ \sf{a = 19 - 8d}} \sf \: ...(i) \\ \\$ And.. $\\$ $\colon \longrightarrow \sf \: 4 {}^{th} \: term + {7}^{th} \: term = 24 \\ \\$ $\colon \longrightarrow \sf \: a + 3d + a + 6d = 24 \\ \\$ $\colon \longrightarrow \sf \: 2a + 9d = 24 \\ \\$ From eq(i) , a = 19 - 8d $\\$ $\colon \longrightarrow \sf \: 2(19 - 8d) + 9d = 24 \\ \\$ $\colon \longrightarrow \sf \: 38 - 16d + 9d = 24 \\ \\$ $\colon \longrightarrow \sf \: - 7d = 24 - 38 \\ \\$ $\colon \longrightarrow \sf \: \cancel{ - 7d }= \cancel{ - 14} \\ \\$ $\colon \implies \underline{ \boxed{ \bf{ \red{d = 2}}}} \: \bigstar$ $\\$ Put it in eq(i) $\\ \colon \longrightarrow \sf \: a = 19 - 8 \times 2 \\ \\$ $\colon \implies \: \underline{ \boxed{ \bf{ \purple{a = 3}}}} \: \bigstar$ $\\$ $\underline{\dag\:\bf Arithmetic \:Progression :-}$ $\\$ » Arithmetic Progression is in the form : ✒ a , a + d , a + 2d , a + 3d ,... $\\$ » For this question, a = 3 d = 2 So, Required Arithmetic Progression would be : $\\$ $\sf{ 3,3+2, 3+2\times2, 3+3\times2,...}$ $\\$ $\therefore\underline{\sf A. P. = \pink{3, 5,7,9,...}}$ $\\$ _____________________________