Question

The sum of the first three terms of an Arithmetic progression (A.P) is 42 and the product of the first and third term is 52 . Find the first term and the common difference.
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Answer 1

Step-by-step explanation:

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$\bf{\bigstar}$$\text{\Large\underline{\bf{Question:-}}}$

$\leadsto$ The sum of the first three terms of an Arithmetic progression (A.P) is 42 and the product of the first and third term is 52 . Find the first term and the common difference.

$\bf{\bigstar}$$\text{\Large\underline{\bf{Solution:-}}}$

$\leadsto$ ● Let a - d , a , a + d are three consecutive terms in arithmetic progression.

$\bf{\bigstar}$$\text{\large\underline{\bf{According\ to\ the\ question:-}}}$

$\leadsto$ (a - d) + a + (a + d) = 42

$\leadsto$ or, 3a = 42

$\leadsto$ or, a = 14....................................(1)

$\leadsto$ again, (a - d) × (a + d) = 52

$\leadsto$ or, a² - d² = 52

$\leadsto$ or, (14)² - d² = 52

$\leadsto$ or, 196 - d² = 52

$\leadsto$ or, d² = 196 - 52 = 144

$\leadsto$ or, d = $\mathrm{\pm}$ 12

$\leadsto$ Hence, first term is (a - d) = 2 [when d = 12] and 26 [when d = -12]

$\bf{\bigstar}$$\text{\Large\underline{\bf{So\ first\ term:-}}}$

$\leadsto$ a = 2 when common difference d, = 12.

$\leadsto$ a = 26 when common difference d, = -12.

$\text{\huge\underline{\bf{Hence\ Verified}}}$

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Answer 2

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Question\;:-}}}$

Here the Concept of A.P. and Missing Terms has been used. We see here first three terms of an AP has been used. We can any three terms in this. We will try to take simplest term in order to solve this. First we will apply the values in case I and then case II. From this, we can find the required values.

Let's do it !!

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★ Equations Used :-

$\\\;\boxed{\sf{(a\;-\;d)\;+\;a\;+\;(a\;+\;d)\;=\;\bf{42}}}$

$\\\;\boxed{\sf{(a\;-\;d)(a\;+\;d)\;=\;\bf{52}}}$

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★ Solution :-

Given,

» Sum of first three terms of A.P. = 42

» Product of first and third term of A.P. = 52

• Let the first term of the A.P. be 'a' and the common difference of A.P. be 'd'.

Then,

• Let the three terms of the A.P. be '(a - d)', 'a' and '(a + d)'.

These three terms are in A.P. only because their common difference is same.

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~ For the First Case ::

Its given that, sum of these three terms of the A.P. is 42. Then,

$\\\;\;\sf{:\Longrightarrow\;\;(a\;-\;d)\;+\;a\;+\;(a\;+\;d)\;=\;\bf{42}}$

$\\\;\;\sf{:\Longrightarrow\;\;a\;-\;d\;+\;a\;+\;a\;+\;d\;=\;\bf{42}}$

$\\\;\;\sf{:\Longrightarrow\;\;3a\;+\;d\;-\;d\;=\;\bf{42}}$

$\\\;\;\sf{:\Longrightarrow\;\;3a\;\;=\;\bf{42}}$

$\\\;\;\sf{:\Longrightarrow\;\;a\;\;=\;\bf{\dfrac{42}{3}}}$

$\\\;\;\underline{\underline{\bf{:\Longrightarrow\;\;a\;\;=\;\bf{14}}}}$

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~ For the Second Case ::

Its given that, the product of Third and First term of the A.P. is equal to 52. Then,

$\\\;\;\sf{:\Longrightarrow\;\;(a\;-\;d)(a\;+\;d)\;=\;\bf{52}}$

$\\\;\;\sf{:\Longrightarrow\;\;a^{2}\;+\;ad\;-\;ad\;-\;d^{2}\;=\;\bf{52}}$

$\\\;\;\sf{:\Longrightarrow\;\;a^{2}\;-\;d^{2}\;=\;\bf{52}}$

Now by applying the value of a = 14 here, we get,

$\\\;\;\sf{:\Longrightarrow\;\;(14)^{2}\;-\;d^{2}\;=\;\bf{52}}$

$\\\;\;\sf{:\Longrightarrow\;\;196\;-\;d^{2}\;=\;\bf{52}}$

$\\\;\;\sf{:\Longrightarrow\;\;\;-\;d^{2}\;=\;\bf{52\;-\;196}}$

$\\\;\;\sf{:\Longrightarrow\;\;\;\cancel{-}\;d^{2}\;=\;\bf{\cancel{-}\;144}}$

$\\\;\;\sf{:\Longrightarrow\;\;\;d^{2}\;=\;\bf{144}}$

$\\\;\;\sf{:\Longrightarrow\;\;\;d\;=\;\bf{\sqrt{144}}}$

$\\\;\;\underline{\underline{\bf{:\Longrightarrow\;\;\;d\;=\;\bf{\pm\;\;12}}}}$

$\\\;\underline{\boxed{\tt{Hence,\;\;common\;\;difference\;\;of\;\;A.P.\;=\;\bf{\pm\;\;12}}}}$

Now applying value of d in the terms we get,

✒ (a + d) = (14 + 12) = 26

Or,

✒ (a + d) = (14 - 12) = 2

$\\\;\underline{\boxed{\tt{Hence,\;\;first\;\;term\;\;of\;\;A.P.\;=\;\bf{26\;\;\tt{or}\;\;\bf{2}}}}}$

>> When d = 12 then a = 26

>> When d = -12 then a = 2

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★ More to know :-

• Verification ::

We already got the value of a and d. By applying these, we get the terms as,

✒ (a - d) = (14 - 12) = 2

✒ (a) = 14

✒ (a + d) = (14 + 12) = 26

Now these are in A.P., since

→ a - (a - d) = (a + d) - a = d

→ a - a + d = a + d - a = 12

→ 12 = 12 = 12

Clearly, the common difference = 12 , hence all terms are in A.P.

Now applying in cases, we get

→ (a - d) + a + (a + d) = 42

→ 2 + 14 + 26 = 42

→ 42 = 42

Clearly, LHS = RHS

Also,

→ (a - d)(a + d) = 52

→ 2 × 26 = 52

→ 52 = 52

Clearly, LHS = RHS.

Here all the conditions satisfy, so our answer is correct. Hence, Verified.

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★ More Formulas to Know :-

$\\\;\sf{\leadsto\;\;a_{n}\;=\;a\;+\;(n\;-\;1)d}$

$\\\;\sf{\leadsto\;\;S_{n}\;=\;\dfrac{n}{2}\;[2a\;+\;(n\;-\;1)d]}$

$\\\;\sf{\leadsto\;\;S_{n}\;=\;\dfrac{n}{2}\;[a\;+\;a_{n}]}$

$\\\;\sf{\leadsto\;\;Last\;Term,\;L\;=\;a\;+\;(n\;-\;1)(-\:d)}$

$\\\;\sf{\leadsto\;\;Arithmetic\;Mean,\;A\;=\;\dfrac{a_{1}\;+\;a_{2}\;+\;....\;+\;a_{n}}{n}}$