Question

The first and the last term of an A.P. are 4 and 304 respectively and sum of n terms of the A.P. is 15554. Find the number of terms and common
difference.​

Answer 1

GIVEN:

• The first and the last term of an A.P. are 4 and 304 respectively.
• Sum of n terms of the A.P. is 15554.

TO FIND:

• What is the common difference and number of terms ?

SOLUTION:

Let the number of terms of an AP be 'n'

To find the number of terms of an AP, we use the formula:-

❰ Sn = $\bf{\dfrac{n}{2}}$ [a + l] ❱

According to question:-

➠ 15554 = $\sf{\dfrac{n}{2}}$ $\times$ [4 + 304]

➠ 15554 = $\sf{\dfrac{n}{2}}$ $\times$ 308

➠ 15554 = n $\times$ 154

➠ $\sf{\cancel\dfrac{15554}{154}}$ = n

❮ 101 = n ❯

Let the common difference of an AP be 'd'

To find the common difference, we use the formula:-

❰ l = a + (n –1)d ❱

According to question:-

➠ 304 = 4 + (101 –1)d

➠ 304 –4 = 100d

➠ 300 = 100d

➠ $\sf{\cancel\dfrac{300}{100}}$ = d

❮ 3 = d ❯

❝ Hence, the number of terms and common difference of an AP are 101 and 3 respectively. ❞

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Answer 2

${ \bold{ \underline{ \sf{Notations \: used:-}}}}$

• First term of A. P = a
• Last term of A. P = l
• Number of terms in entire A. P = n
• Common difference = d
• Sum of A. P = s

${ \bold{ \underline{ \sf{Given:-}}}}$

• a = 4
• l = 304
• s = 15554

${ \bold{ \underline{ \sf{To \: Find:-}}}}$

• n = ?
• d = ?

${ \bold{ \sf{ \underline{Solution:-}}}}$

${ \boxed{ \bf{s = \dfrac{n}{2} \times (a + l)}}}$

By putting the values in above formula we get

$\implies{ \bf{15554 = \frac{n}{2} \times (4 + 304)}}$

$\implies{ \bf{15554 = \frac{n}{2} \times 308}}$

$\implies{ \bf{15554 = n \times 152}}$

$\implies { \bf{ \dfrac{15554}{152}}}$

${ \boxed{ \red{ \bf{ \implies \: n = 101}}}}$

Now,

${ \boxed{ \bf{i = a + (n - 1) \times d}}}$

$\implies{ \bf{304 = 4 + (101 - 1)d}}$

$\implies{ \bf{304 - 4 = 100d}}$

$\implies{ \bf{ \dfrac{300}{100}}}$

${ \boxed{ \red{ \bf{ \implies \: d = 3}}}}$

Therefore Number of terms in A. P is 101 and common difference is 3 .

NOTE:-

Such series is not possible because difference(d) and number of terms(n) are not integer values.

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