Question

The sum of first 7 term of an AP is 63 and the sum of its next 7 term is 161. find the 28th term of this AP

Answer 1

Find the answer in attachment.

Answer 2

$\large{\underline{\bf{\pink{Answer:-}}}}$

✰ The 28th term of the given Ap is 57.

$\large{\underline{\bf{\blue{Explanation:-}}}}$

$\bf\:S_n=\frac{n}{2}[2a+(n -1)d]$

where,

• sn = sum of n terms of an AP .
• a = first term
• n = number of terms
• d = common difference

$\large{\underline{\bf{\green{Given:-}}}}$

✰ sum of first 7 term =63

✰ sum of next 7 term = 161

$\large{\underline{\bf{\green{To\:Find:-}}}}$

✰ we need to find the 28th term of AP.

$\huge{\underline{\bf{\red{Solution:-}}}}$

Let a be the first term and d be the common difference of the given AP.

Then,

using $\bf\:S_n=\frac{n}{2}[2a+(n -1)d]$

we get,

$:\implies\bf\:S_7=\frac{7}{2}[2a+6d]$

$:\implies$ ⠀⠀⠀7 (a + 3d) = 63

$:\implies$ ⠀⠀⠀a + 3d = 9............(1)

Clearly, the sum of first 14 terms = 63+161 = 224.

So,

s14 = 224

$:\implies$ ⠀$=\frac{14}{2}[2a+13d]=224$

$:\implies$ ⠀⠀⠀7(2a + 13d) = 224

$:\implies$ ⠀⠀⠀2a + 13d = 32...........(2)

Multiplying (i) by 2 and substracting the result from (ii),

we get

$:\implies$ ⠀⠀⠀7d = 14

$:\implies$ ⠀⠀⠀d = 2

putting d = 2 in (i) ,we get a = 9-6 = 3.

thus, a = 3 and d = 2

So , the 28th term of this AP is given by

T28 = (a + 27d)

$:\implies$ ⠀⠀⠀(3 + 27 × 2)

⠀⠀⠀ ⠀⠀⠀ ⠀⠀⠀ ⠀⠀⠀ ⠀= 57

Hence , the 28th term of the given AP is 57.

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